an exploratory study into geometry understanding ishaak
TRANSCRIPT
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An exploratory study into grade 12 learners’ understanding of
Euclidean Geometry with special emphasis on cyclic quadrilateral
and tangent theorems
Ishaak Cassim
A research report submitted to the School of Science Education at
the University of the Witwatersrand, Johannesburg, in partial
fulfilment of the requirements for the degree Master of Science.
October 2006
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Declaration
I, Ishaak Cassim, declare that this research report is
my own, unaided work. It is being submitted for the
degree Master of Science at the University of the
Witwatersrand, Johannesburg. The research has not
been submitted before for degree or examination at
any other University.
________________
Signature of Candidate
______________________day of_________________2006.
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Abstract
This research report explored the strategies which grade 12 learners employ to solve
geometric problems. The purpose of this research was to gain an understanding of
how grade 12 learners begin to solve geometric problems involving cyclic
quadrilateral and tangent theorems. A case study method was used as the main
research method. The study employed the van Hiele level’s of geometric thought as a
method for categorising learners levels of understanding. Data about the strategies
which learners recruit to solve geometric problems were gathered using learner-based
tasks, semi-structured interviews and document analysis.
From the data gathered, the following patterns emerged: learners incorrect use of
theorems to solve geometrical problems; learners base their responses on the visual
appearance of the diagram; learners “force “ a solution when one is not available;
learners’ views of proof. Each of these aspects is discussed.
The report concludes that learners strategies to solving geometric problems are based
largely on the manner in which educators approach the solving of geometrical
problems.
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Acknowledgements
My sincere thanks to all involved in the completion of this research
report and in particular the following:
• My Creator for the strength and guidance He has provided me-
• My supervisor, Dr Willy Mwakapenda for his constant and
professional advice and support throughout the study;
• My parents (Mogamat and Mirriam Cassim), brother (Abdul
Rafick Cassim), sisters (Yasmeen Abdulla and Aneesa Cassim),
wife (Fatima Bibi Cassim) and son (Muhammed Yusuf Cassim)
for their support and encouragement to ensure this study is
completed successfully;
• Mr M H Allimia and Mr M L Nkwane for proof reading and
editing my report;
• The Principal, educator and learners of the school at which the
study was undertaken;
• Finally, to all those not mentioned, especially my colleagues at
work thank you for the encouragement and support you
provided to me during this study.
I.Cassim
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List of Tables Table 1: A summary of van Hiele’s model of geometric reasoning. ......................... 27 Table 2: The van Hiele model of thinking together with the phases of learning........ 32 Table 3: Terms used at levels 2 and 3 of the van Hiele model .................................. 33 Table 4 : Performance of Higher Grade grade 12 learners in Paper 2 (ex HOD , 1993: 1) ............................................................................................................................ 43 Table 5: Standard Grade learners average percentage in different sections of Paper 2 (ex HOD , 1993 : 1)................................................................................................. 43 Table 6:Standard Grade learners performance in some questions of the question paper (Maths Standard Grade Paper 2) (MST, 2003:6) ................................................... 44 Table 7: HG learners performance in Mathematics 2002 Grade 12 question paper. 45 Table 8: The preferences of qualitative researchers (Silverman, 2000:8).................. 49 Table 9: Comparison between the pilot and main study. ..........................................65 Table 10: Categories of learner’s responses. ............................................................ 78 Table 11: Learners responses according to categories outlined in Table 10 .............. 79 Table 12: Learner’s response to tasks in Section A .................................................. 80 Table 13: A summary of learners’ responses to Question 1...................................... 81 Table 14: L24’s response to Question 1 ................................................................... 81 Table 15: A summary of learners’ response to Question 2 ....................................... 85 Table 16: L 8’s response to Question 2 of Section A................................................ 86 Table 17: Summary of learners’ responses to Question 3 ......................................... 87 Table 18: Summary of learners’ response to Question 4........................................... 89 Table 19: The proof provided by L12 to Question 4................................................. 90 Table 20: A response to Question 4 by L2 ............................................................... 91 Table 21: Summary of learners’ responses to Question 5 ......................................... 92 Table 22: A response to Question 5 by L8 ............................................................... 93 Table 23: Summary of learners’ responses to Question 6 ......................................... 93 Table 24: Summary of learners’ responses to Question 7 ......................................... 95 Table 25: L 3’s response to 1.3 of Section B............................................................ 97 Table 26: L10’s response to 1.3 of Section B........................................................... 98 Table 27: L4’s response to 1.3 of Section B............................................................. 99 Table 28: A learner’s response to illustrate the phenomenon of “distortion of.........101 theorems” ...............................................................................................................101 Table 29: Key aspects according to which learner’s reasoning skills were catagorised................................................................................................................................108
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List of Figures Figure 1: Typical geometry problem involving tangent-chord theorem and a learner’s response to such a problem...................................................................................... 11 Figure 2: Squares of different sizes.......................................................................... 24 Figure 3: Rectangles of different sizes……………………………………………….24 Figure 4: An example of how learners’ knowledge from different section of the Mathematics can be recruited to solve a given problem........................................... 31 Figure 5: A typical proof which grade 12 learners are expected to reproduce under test / examination conditions.......................................................................................... 37 Figure 6: A typical proof which is presented to Grade 11 learners’ as a finished product. ................................................................................................................... 38 Figure 7: Graphic representation of triangulation process………………………… 52 Figure 8 : Flowchart for interviews as a research instruments (Chetty,2003:41) ....... 60 Figure 9: Learner 24’s diagram used. Note markings on the diagram which informed her choice of answer................................................................................................ 83 Figure 10 : The diagram used by L2 to answer Question 1. Note the markings used on the diagram. ............................................................................................................ 84 Figure 11: The diagram L 6’s used to solve Question 3. Note the markings on the diagram made by the learner.................................................................................... 88 Figure12: A typical geometric rider which grade 12 learners’ should be able to solve................................................................................................................................117 Figure 13: A schematic representation demonstrating the method of “backward reasoning”. .............................................................................................................118
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List of Abbreviations Used AMESA Association of Mathematics Education of South Africa
DoE Department of Education
FET Further Education and Training
GDE Gauteng Department of Education
HOD House of Delegates
MASA Mathematical Association of South Africa
MST Mathematics, Science and Technology
NCS National Curriculum Statements
RNCS Revised National Curriculum Statement
TED Transvaal Education Department
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Table of Contents
CHAPTER 1: INTRODUCTION..................................................................................................10
1.1 GENERAL INTRODUCTION.................................................................................................10 1.2 THE PROBLEM TO BE INVESTIGATED.......................................................................................10 1.3 FOCUS OF THE STUDY..............................................................................................................16 1.4 RESEARCH QUESTIONS............................................................................................................16 1. 5 RESEARCH METHODS.............................................................................................................17 1.6 ANALYSIS OF DATA .................................................................................................................18
1.6.1 Video Recording of Lesson..............................................................................................18 1.6.2 Interviews .......................................................................................................................18 1.6.3 Learner- Based Task .......................................................................................................19
1.7 LIMITATIONS OF THE STUDY ....................................................................................................19 1.8 ORGANISATION OF THE REPORT...............................................................................................20
CHAPTER 2:..................................................................................................................................22
THEORETICAL FRAMEWORK AND LITERATURE REVIEW .............................................22
2.1 THEORETICAL FRAMEWORK ....................................................................................................22 2.1.1. Theoretical Framework: Introduction.............................................................................22 2.1.2 The Van Hiele Model Of The Development Of Geometric Thinking..................................23
2.1.3 THE MODEL EXPLAINED................................................................................................24 Level 0: Visualisation ..............................................................................................................24 Level 1: Analysis .....................................................................................................................25 Level 2: Informal Deduction ....................................................................................................25 Level 3: Formal Deduction......................................................................................................25 Level 4: Rigour........................................................................................................................26 2.1.4 How does the model work................................................................................................27 2.1.5 Learning Phases ........................................................................................................29 2.1.6 The role of language .......................................................................................................32 2.1.7 Conclusion......................................................................................................................34
2.2. LITERATURE REVIEW .............................................................................................................35 2.2.1 Introduction....................................................................................................................35 2.2.2 The problem about Curriculum .......................................................................................36 2.2.3 The Performance problem...............................................................................................42
CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY.......... .........................................48
3.1. THE RESEARCH DESIGN..........................................................................................................48 3.2 ACCESS TO PARTICIPANTS.......................................................................................................50
3.2.1 Data collection instruments.............................................................................................52 3.2.2 Interviews .......................................................................................................................53 3.2.3 Summary of advice about designing and using interviews ................................................57
3.2.4 DOCUMENT ANALYSIS ..........................................................................................................63 3.2.5 PARTICIPANT OBSERVATION OF LESSONS...............................................................................63
CHAPTER 4: DATA ANALYSIS AND FINDINGS .....................................................................78
4.1 ANALYSIS OF LEARNERS’ RESPONSES.......................................................................................78 Coding process........................................................................................................................78 Question by question analysis of learners’ responses ...............................................................80
CHAPTER 5: DISCUSSION OF FINDINGS.............................................................................. 100
5.1 HOW LEARNERS’ USE THEOREMS TO SOLVE A GIVEN PROBLEM. ............................................... 100 5.2 LEARNERS RESPONSES ARE BASED ON THE VISUAL APPEARANCE............................................. 102 OF A GIVEN FIGURE..................................................................................................................... 102 5.3 LEARNERS “FORCE” A SOLUTION TO A GIVEN TASK ALTHOUGH NO.......................................... 104 SOLUTION IS POSSIBLE AT TIMES.................................................................................................. 104 5.4 LEARNERS’ VIEWS OF PROOF FOR A GEOMETRICAL PROBLEMERROR! BOOKMARK NOT DEFINED .
CHAPTER 6: SUMMARY, RECOMMENDATIONS AND CONCLUSION . .......................... 107
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6.1 SUMMARY AND FINDINGS ......................................................................................................107 6.1.1 Overview ...................................................................................................................... 107 6.1.2 Primary research questions and sub-questions .............................................................. 108
6.2 IMPLICATIONS AND RECOMMENDATIONS FOR CLASSROOM PRACTICE....................................... 111 How to teach Geometry ......................................................................................................... 111 Teach to prove or teach for proof?......................................................................................... 114 Use of interactive technology................................................................................................. 119
6.3 CONCLUSION ........................................................................................................................ 120
BIBLIOGRAPHY......................................................................................................................... 123
APPENDIX 1 : LETTER OF PERMISSION TO CONDUCT RESEARC H AT SCHOOL....... 128
APPENDIX 2: LEARNER-BASED TASKS................................................................................ 129
APPENDIX 3: LEARNERS’ RESPONSES TO TASKS............................................................. 130
APPENDIX 4: EXAMINERS’ REPORTS................................................................................... 131
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Chapter 1: Introduction
1.1 General Introduction
This study is an investigation into grade 12 learners’ understanding of Euclidean
Geometry at one school in Gauteng, South Africa. In this chapter, I will discuss the
problem to be investigated, the purpose of the study, the research questions
investigated, and the site of the study. An outline of the research methods and data
analysis is also included.
This study investigated an understanding of Euclidean Geometry with specific
reference to cyclic quadrilaterals and tangent theorems of a group of grade 12
learners’ at an independent school. The order of the discussions outlined is to provide
a logical argument on the relevance and importance of this study for mathematics
teachers teaching Euclidean geometry at grades 10 – 12 levels.
1.2 The problem to be investigated
When compared with other school mathematics content areas, the topics covered in
Euclidean Geometry have remained constant. An analysis of the Interim Core
Syllabus (Department of Education (DoE), 2003) is testimony to this. Learners’
especially, grade 12’s, performance in school geometry has also been reported to be
inadequate. Examiners Reports (House of Delegates (HoD); Gauteng Department of
Education (GDE) 1995, 2001, 2002, 2003) as well as the Mathematics, Science and
Technology (MST) Report (2003), all comment on learners poor performance in
Euclidean Geometry, indicating the following as some of the aspects that are typical
of the way that learners respond to exam questions in Euclidean Geometry:
When proving theorems, learners omit necessary constructions from their diagrams
and statements are not written in a logical sequence. Learners often base their
responses to a question on the visual appearance of a given diagram, resulting in
learners making assumptions not directly related to the given diagram. When asked to
calculate the magnitude (size) of angles, learners’ often assign specific values to the
measures of angles. The concepts of similarity and congruency are often confused
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with one another. Learners have difficulty in identifying the exterior angles of
triangles or cyclic quadrilaterals, when they do appear in diagrams. A common
problem is the identification of the angle between the tangent and chord and the angle
in the alternate segment. The figure below illustrates this typical problem.
Figure 1: Typical geometry problem involving tangent-chord theorem and a learner’s response to such a problem.
In the above proof, the learner incorrectly identifies1
∧Y , as the angle in the alternate
segment in relation to1T∧
. Learners’ poor performance in Geometry is not only limited to South Africa. The
scope of most writings on Euclidean Geometry focuses on the twin aspects of learners
“poor performance of students and an outdated curriculum” (Usiskin, 1987: 17). In an
attempt to provide an explanation for learner’s poor performance in geometry, Usiskin
(1987), cites Allendoefer (1969) who writes:
“The mathematical curriculum in our elementary and secondary school faces a
serious dilemma when it comes to geometry. It is easy to find fault with the
traditional course in geometry, but sound advice on how to remedy these
difficulties are hard to come by ……. Curricular reform groups at home and
abroad have tackled the problem, but with singular lack of success or agreement
… …. We are, therefore under pressure to “do something” about geometry; but
what shall we do?” (Usiskin; 1987 : 17).
To echo Allendoefer (1969), the question to ask is :“What shall we do” to
improve learners’ performance in Euclidean Geometry?
In the example, on the right, KT is a tangent to the circle at T.The chord NM is produced to meet the tangent KT at K.Y is a point on the chord NM. The proof below highlights a learner’s response:
24
11
21
YT
YT
TTN
∧∧
∧∧
∧∧∧
=
=
+=
(Transvaal Education Department (TED), 1994:7)
T
N
Y
M
K
4
32
1
21
21
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Why have I decided to focus on Geometry? Some of my reasons are cited below:
• Through school visits as a Subject Advisor for Mathematics, in two different
areas of Gauteng province, I have observed that both primary and secondary
school educators tend to delay the teaching of Geometry to as late as possible
in the school year.
“Teaching geometry is very often left to the last term of the year where a few lessons are
taught because this is what is being assessed” ( Penlington, 2004 : 192).
• The poor performance of grade 12 learners’ in geometry as compared to other
aspects (sections) of mathematics.
• The present Interim Core Syllabus (GDE, 1995), which is used in grades 10 –
12, does little to advance the improved teaching of Geometry. Educators are
provided with a syllabus, which is prescriptive in nature, which enumerates
the theorems and their converses that need to be covered in a particular grade.
The document lacks instructional (pedagogical) methodology of the teaching
of the required concepts. Often learners are instructed by teachers to memorise
proof of theorems or properties of geometric figures. A pre-service education
student at Wits University, succinctly captures this scenario when he writes in
his journal:
“(At school) we were given properties to learn by heart and never knew
for sure how true is (it) that, for example ( the) exterior angle of the
triangle is equal to opposite interior angles” (Pournara, 2004 : 208).
Current textbooks do little in developing the learner’s ability to “develop
insight into spatial relationships and measurements” (DOE, 1995). They
proceed directly to the formal proof of the theorem, without first allowing the
learners to get a “feel” of what the theorem is all about.
• The current specific outcomes for Mathematical Literacy Mathematics and
Mathematical Sciences (MLMMS) as stated in the policy document for the
Intermediate and Senior Phase (DOE, 1997) do not provide clear, lucid,
transparent instructions as to what specific content needs to be covered.
Specific Outcome 7, dealing with Geometry (space, shape), reads : “ Describe
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and represent experiences with shape, space, time and motion, using all
available senses.” (DOE, 1997). In the primary grades (i.e. grades 4-9), the
emphasis in Geometry is the development of spatial senses, which is important
for later studies in Geometry. Van Niekerk (1998) contends that for learners to
develop their spatial senses, educators should start with learners ordering and
structuring their spatial experiences, which they can encounter in their
everyday experiences – which are primarily a three-dimensional experience.
This has implications for instructional resources – paper, pencil, scissors, pritt
(glue), compasses, etc, which would be required to undertake the activities
such as the construction of models. The discrepancies in resourcing of our
schools mean that if learners do not have access to these “tools” they may not
develop the required skills adequately. Once this basic skill (of visualization)
has been adequately developed, it would then be appropriate to lead learners to
the more structured and formal geometry encountered at grades 10 – 12 level.
“National as well as international research has shown that the majority of
learners in schools tend to have a backlog in their intuitive understanding of
space in comparison with their intuitive number knowledge” (Van Niekerk,
1998: 70).
• The Revised National Curriculum Statement (RNCS) (DoE, 2002), emphasise
a hands – on, practical approach to geometry in the grades R-9 band. “The
study of geometry requires thinking and doing” (Penlington, 2004: 192). The
learner moves from low level (according to Van Hiele’s model) of recognition
and description skills to higher order skills of classification and discrimination
of two-dimensional objects. In skills such as construction of models, different
views of objects are used to be developed in the learner during this phase (in
grades 4 – 9).
• In the National Curriculum Statement (NCS), ( DoE, 2003) for school for
grades 10 -12, learning outcome 3 deals specifically with space, shape and
measurement. Learners’ are exposed to an enquiring, investigative,
developmental approach to Geometry. Learners’ are encouraged to investigate,
conjecture and discover through guided learning experiences. The
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relationships between, for example, angles and the sides opposite then in any
triangle. They are encouraged to test the validity of their conjectures using an
array of resources, including computer software like Geometers’ Sketchpad
and Cabri, for example.
From the preceeding discussion one can trace changes in the approach to how
Geometry is to be taught at school level, as well as the skills, knowledge that learners
are expected to gain access to. However, the current grade 10 – 12 learners in the
system will be assessed using a method that is to be discontinued in 2008. So
educators are caught in a “Catch 22” situation: whilst they may be allowed (even
encouraged) to use the new methods related to investigations, and discovery, which
requires time, and places other demands on both educator and learners, learners are
expected to be assessed using methods which do not reflect this type of learning. The
positive aspect of this approach, however, is that the learner’s understanding will be
enriched if the method is appropriately applied.
The purpose of this study is to gain insights into grade 12 learners’ understanding of
Euclidean Geometry. I have specifically selected grade 12 learners because they
would have received at least (8) eight years exposure to Euclidean Geometry.
Whilst other researchers such as, De Villiers, Lubisi and Mudaly, (South African
Math educationists), have explored and written extensively on the nature and purpose
of Euclidean Geometry, none of these have succinctly explored learners’
understanding Euclidean Geometry. Furthermore, the Van Hiele levels of Geometric
thinking which is used as a lens through which I undertook this study is restricted to
rectilinear shapes like squares and rectangles. It has not been used to investigate
learners’ understanding of Geometry which involves cyclic quadrilaterals and tangent
theorems. The van Hiele model was selected as it has not been designed specifically
for Euclidean Geometry only, “but also identifies a way in which the level of
geometry argumentation or thinking can be measured” (van der Sandt & Nieuwoudt,
2004:251).
In the South African context, the study undertaken by Human, Nel, De Villiers,
Dreyer and Wessels (De Villiers, 1997), which undertook to rethink the manner in
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which Euclidean Geometry was taught in South African Schools, whilst important did
not seem to take flight, what I mean is that, with the exception of the participating
schools, their findings regarding the manner in which the geometry ought to be taught
at secondary school level was certified ‘dead on arrival’.
In the Human et al (1977) study, (De Villiers, 1997) cited amongst others the
following change to the way in which geometry is taught in the South African context
is explained:
• Informal geometry as it has been presented in South African schools till the
end of grade 9 (std 7).
• Proof restricted to propositions, which really require justification and / or
explanation, and assuming without proofs all propositions about which pupils
have no doubt.
• Construction of formal, economical definitions for the different types of
quadrilaterals, and logically deducing the other properties from the definitions
as a first exercise in local axiomatizing
• Local axiomatizing regarding other groups of related propositions, for
example, their related to intersecting and parallel lines.
• Global axiomatizing ( not included in the experimental course ).
(De Villiers, 1997: 37)
Almost thirty years later, and as a result of change in political leadership in South
Africa, one does notice a minute, though significant change to the geometry
curriculum to be offered in South African schools.
The thrust of this study, affords one a snap view of learners’ understanding of
Euclidean Geometry. This understanding is framed to a large extent in the manner it is
taught to learners, i.e. as a set of theorems, which need to be memorized for the final
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grade 12 exams. This study, though limited in its scope, aims to make a meaningful
contribution to the pedagogical knowledge of practicing mathematics educators for
improving learners’ understanding of Euclidean Geometry.
1.3 Focus of the study
The discipline of mathematics, and Euclidean Geometry in particular, offers to the
researcher an almost seamless avenue of research prospects. However, time and space
limit the focus and scope of this study. The focus of this study was on the learners.
However, teacher practice is also considered, but in a very limited way. The research
conducted in this study documented learners understanding of Euclidean Geometry
with specific reference to cyclic quadrilaterals and tangent-chord theorems.
Whilst learners in secondary schools, i.e. from grade 8 – 12 are exposed to formal
geometry this study focuses on grade 12 in particular. The study is further delimited
in that it involved only one school from Tshwane South District.
My familiarity with schools and maths educators was significant, as the participating
school was selected on the basis of their excellent grade 12 results over the past five
years. The selection of grade 12 learners was significant, as it is the culminating point
of 12 years of formal schooling as well as at least 5 years exposure to formal
geometry. This research would benefit mathematics teachers in the lower grades (i.e.
grade 8 – 11) as well as curriculum planners.
The intention of this study is not to generalize findings, but to ensure that the findings
are relatable to other grade 12 learners and educators working at this level of the
school curriculum.
1.4 Research Questions
An exploratory study of one grade 12 educator and, her group of grade 12 learners in
the Tshwane South District in Gauteng, attempted to answer the following research
questions:
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1. How do grade 12 learners begin to solve a geometric problem?
2. What knowledge and skills do learners employ in order to prove
geometric problems?
The above questions were the focus of this study: to explore and understand how
grade 12 learners understand Euclidean Geometry. The aim of the study was to
enquire, explore, interpret, understand and report on grade 12 learners understanding
of Euclidean Geometry, with specific reference to cyclic quadrilateral and tangent-
chord theorems.
1. 5 Research Methods
The research method adopted for this study was an exploratory study in grade 12
learners’ understanding of Euclidean Geometry with specific reference to cyclic
quadrilaterals and tangent chord theorems. Aspects relating to research method, i.e.
the research design, suitability to a qualitative framework, development and reliability
of data collection tools are set out in detail in chapter 3. For brevity I used three data
collection tools:
1. Interviews.
2. Lesson observation (video recording).
3. Learner -based tasks.
The reliability of the instruments and the triangulation of data emerging from them
are key themes that feature in subsequent chapters. The interviews and learner tasks
were piloted in order to improve and determine:
1. Their user friendliness to participants;
2. Understanding of key terms;
3. Ensuring relevant generation of data;
4. Analysis and interpretation of data gathered;
5. Researchers’ skills in conducting interviews;
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6. Validity of instruments and
7. Reliability of findings.
1.6 Analysis of Data
1.6.1 Video Recording of Lesson
The Educator’s lesson was recorded to establish her method of teaching geometry.
Very often learners mimic the way they are taught a section when working on their
own or in groups on an item relevant to the topic. Adhering to the qualitative research
paradigm, an open coding system was employed. Fraenkel and Wallen (1993)
maintain that open coding takes place when data collected is examined for patterns
and / or categories, which are then later further narrowed down.
1.6.2 Interviews
Interview schedules were used to conduct interviews with five learners from the grade
12 class. The interviews generated data on :
• Learners’ understanding of the task posed
• The strategies learners’ employ when solving geometric problems
(tasks).
Furthermore, audiotapes of the learners interviews were employed to prepare the
interview transcripts. The questions posed to the learners in the interview were open –
ended in order to obtain a rich, thick description of learners’ understanding of
Euclidean Geometry and the tools they employ to solve geometric problems. Like the
video recording of the teacher’s lesson, the interviews transcripts were also analysed
using an open coding system.
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1.6.3 Learner- Based Tasks
The learners were each asked to complete the attached task on their own (see
Appendix 2). The tasks involved geometry problems involving tangent – chord
theorem and cyclic quadrilateral theorem. The tasks were sourced from Daly (1995).
An open coding system was utilised to develop the categories of learner responses.
According to Fraenkel and Wallen (1993), a major benefit regarding the usage of
content or document analysis is that its an unobtrusive tool, through which the
researcher can peep without being present and that which is analysed is not affected
(adversely or otherwise) by the researcher’s presence.
1.7 Limitations of the Study
When research is conducted in the classroom such research is inherently constrained.
Such limitation is often beyond the control of the researcher. The limitations of this
study include the following:
• The problem under investigation focuses on the learner and not on the
teacher;
• A single grade 12 class was selected at a particular school in the
district.
Such a small sample could begin to question the external validity of the findings.
This study was conducted within the paradigm of qualitative research. Marshall and
Rossman (1989), argue that there is a weakness in qualitative research in
transferability of results as each qualitative research approach has its own unique
features. Although much of the data analysed was based on the researcher’s subjective
interpretation, it should be noted that this could result in bias of findings. Fraenkel and
Wallen (1993), argue that no matter how impartial an observer attempts to be, there is
always some element of biasness present.
An audiotape was used to record the interviews with the learners. The presence of the
audiotape did at times affect the sincerity of learners’ responses. However, as the
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interviews progressed, learners’ became more accustomed to the presence of the audio
tape recorder.
In qualitative research the quality of the data collected depends largely on the skills of
the researcher. In qualitative research the social climate is always changing and the
researcher is not always able to account for changing conditions in the phenomenon
being studied. In this study most of the research was undertaken in the third school
term.
In a qualitative research paradigm, it is not always easy to overcome the above
limitations. It is, however, important to acknowledge such limitations and attempt to
minimize their influence on the research process.
1.8 Organisation of the Report
Whilst this chapter has outlined key reasons for the research, further reasons are
advanced and clarified in later chapters. The report constitutes the following parts:
Chapter 1: Introduction and purpose of study.
Chapter 2: The literature review which supports this study as well as the theoretical
framework which forms the basis of the analysis and arguments put forward in this
report.
Chapter 3: A detailed report of the research method, development of materials and
processes undertaken to improve on the reliability of the results. This chapter includes
an in depth discussion of the different data collection tools used.
Chapter 4: Analysis of the research data and key insights that are flagged for
discussion. Issues emanating from the pilot study and key issues from the main study.
.
Chapter 5: Discussion platform for key aspects of the research, which are also
criticised. .
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Chapter 6: Summary of results, recommendations and conclusions based on the one
cohort of grade 12 learners’ understanding of Euclidean Geometry.
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Chapter 2: Theoretical Framework and Literature Review
2.1 Theoretical Framework
2.1.1. Theoretical Framework: Introduction
Traditional Euclidean geometry teaching focused on the formal write up of proofs to
given geometric riders. Whilst most learners performed relatively well in other
branches of Mathematics (like Algebra and Trigonometry) they performed dismally
when it came to Euclidean Geometry. Comments such as:
- “ Had to prove theorems all year long;”
- “ Didn’t understand what it was all about;”
- “I passed geometry by memorizing proofs;”
are indicative of learners’ dissatisfaction for Euclidean Geometry. In a study
undertaken by Pournara (2004), a group of prospective teachers at the University of
the Witwatersrand were asked, “Why do we study geometry at school?” Some of their
responses included:
o To prove theorems;
o Perhaps to bring marks down a bit. (emphasis added)
o I don’t know but our teacher used to say that we will need the skills we
learn in geometry to apply it in our everyday lives
o I don’t know why we study geometry at high school because the vast
majority of it cannot be applied to everyday life, nor does it have
meaning or relevance to the learners’ lives
o Geometry is about using theorems to attain results. (emphasis added)
The comments above reflect a myopic view of geometry, held by prospective teachers
who are expected to teach the subject in the not to distant future. The above comments
demonstrate that learners at school cannot see the utilitarian value of geometry outside
of the school environment.
In recent times there have been suggestions that Euclidean Geometry should be
scrapped from the South African School Curriculum totally. In the July (1996) edition
of the Mathematical Digest, one author wrote, “ South Africa is the habitat of an
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endangered species, for Euclidean Geometry has disappeared from the syllabus of
most other countries” (p. 26). Despite such pronouncements by certain sceptics,
“Geometry is alive and well” (de Villiers, 1996: 37), and experiencing a revival
(renaissance) in many countries throughout the world. There is no doubt about the
importance of geometry not only to develop logical thinking, but also as a support to
developing insights into other branches of Mathematics as well as in fields of study
such as engineering, architecture, physics and astronomy.
What we need is a change of strategy to make geometry more easily understood and
readily appreciated. To this end, then, we need to develop the linguistic register
(vocabulary), concepts, etc, in order to create the necessary insight and understanding
of the deductive system, and in this way recapture the fascination of geometry without
having to memorise proofs. The question to be answered is HOW DO WE DO THIS?
There is a model called the van Hiele model of geometric thinking, which can be used
as a teaching tool as well as an assessment tool.
2.1.2 The Van Hiele Model Of The Development Of Geo metric
Thinking
The van Hiele model of geometric thinking came out as a result of the doctoral
dissertation of the van Hiele couple, Pierre van Hiele and his wife Dina van Hiele –
Geldof, at the Dutch University of Utrecht in 1957. It was sad that Dina died shortly
after completing her dissertation and it was her husband Pierre who advanced the
theory further.
Pierre’s dissertation focused primarily on problems experienced by learners in
geometry education, while Dina’s focused on “teaching experiment” (de Villiers,
1997: 40) i.e. on the sequencing of geometric content and learning activities for
learners. The former Soviet Union in the1960’s was amongst the first countries in the
world to realign her geometry curriculum so that it adhered to the van Hiele model.
Since then, however, the model has gained prominence in most countries.
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Its distinct levels of understanding characterize the model. The five levels of
understanding are labelled from the most basic task to the most cognitively
demanding as:
- Level 0: Visualisation
- Level 1: Analysis
- Level 2: Informal Deduction
- Level 3: Formal Deduction
- Level 4: Rigour
The model contends that through appropriate instruction (teaching) learners’ progress
from the basic level of visualization, where learners only observe shapes on their
physical properties, through to the highest level, which is concerned with “formal
abstract aspects of deduction.” (Crowley 1987:1).
2.1.3 THE MODEL EXPLAINED
Level 0: Visualisation
At this basic level of the model the learner is basically aware of the space around him
/ her. Geometric objects are considered in their totality rather than in terms of their
properties or constituent parts. A learner at this level has the ability to identify
specific shapes, reproduce them and learn the appropriate geometric vocabulary
(Crowley, 1987). For example, a learner at this level may be able to identify that
figure 1(below left) contains squares and figure 2 (below right) contains rectangles.
However, a learner would not be able to state that the opposite sides of a square are
parallel or the angles at the vertices are 90o.
Furthermore, given dotty or square paper the learners would be able to reproduce the
given sketches to some degree of accuracy.
Figure 2: Squares of different sizes. Figure 3: Rectangles of different sizes.
- 25 -
Level 1: Analysis
At the level of analysis properties of geometric shapes are being understood by
learners’ through experimentation and observation. These new properties are used to
conceptualise classes of shapes. For instance a learner is able to recognize that a
square is a rhombus, since a square has all the properties of a rhombus. While learners
at this level are able to master the relevant technical knowledge to describe figures
(shapes), they still lack the capacity to “interrelate figures or properties of figures” (de
Villiers, 1997:41), and make sense of definitions.
Level 2: Informal Deduction
At this level of understanding, learners are able to establish the interrelationships that
exist between and among figures. For instance learners are able to state that in a
quadrilateral, if the opposite sides are parallel, then the opposite angles are equal, as
well as that a square is a rectangle as it has all the properties of a rectangle.
At this level then, learners are able to deduce properties of a figure and also recognize
classes of figures. Learners are able to understand class inclusion. Definitions begin to
make sense for learners and are understood by them. However, at this level, learners
are not able to “comprehend the significance of deduction as a whole or the role of
axioms” (Crowley, 1987:3). Some Mathematics educationists, and some textbook
authors regard axioms as self-evident truths – they do not regard axioms as the initial
building blocks of a mathematical system (de Villiers, 1997). As a result of their
misrepresentation, learners are also informed incorrectly.
Level 3: Formal Deduction
At this level learners are able to understand “the significance of deduction, the role of
axioms, theorems and proof” (de Villiers, 1997: 41). At this level the learners have
the ability to construct proofs based on their own understanding. They do not need to
memorize readymade proofs and produce them on demand in an exam or test. The
learner is able to develop a proof in more than one way. Furthermore, “the interaction
- 26 -
of necessary and sufficient conditions is understood; distinctions between a statement
and its converse can be made” (Crowley, 1987: 3). However, very few learners reach
this stage of “advanced” reasoning.
Level 4: Rigour
The learner at this level does not function at the Euclidean deductive axiomatic
system only. The learner has the potential to study non-Euclidean systems such as
spherical geometry. By being exposed to other axiomatic systems, the learner is able
to compare similarities and differences that exist between the systems. Geometry can
be studied / seen in an abstract form.
Of the five levels of the model this last level is the least developed originally. As most
high school geometry is taught at level 3 it should not be surprising then that most of
the research done focuses on the lower levels of the model.
The key features of the model can be summarized as follows in Table 1.
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Table 1: A summary of van Hiele’s model of geometric reasoning. LEVELS What is studied How are they studied Examples
0 Individual objects e.g.: squares or
rectangles only.
Visually recognized on
the basis of physical
appearance.
Rectangles of all sizes
having same orientation
grouped together on the
basis of their
orientation or
appearance.
1 A class of shapes, e.g. a square is a
rhombus since a square has all the
properties of rhombuses.
Figures having common
characteristics are
grouped together, e.g.
squares are a subclass of
rhombuses.
A square has all
adjacent sides equal.
Diagonals bisect each
other at right angles.
Opposite sides are
parallel.
Opposite angles are
equal, etc.
2 Learner begins to define figures /
shapes belonging to same grouping
(family).
Observing and noticing
relationships between
properties studied. This
is done largely on an
informal basis.
Through measurement
of diagonals of squares
learners will conclude
that they intersect at
right angles and that
they are equal.
3 More formal proofs are studied Using axiomatic system
prove relationships. A
more formal approach is
adopted.
Prove formally that a
square is indeed a
rhombus.
4 Geometry is studied on an abstract
level. There is a move between
systems (e.g. using algebraic system to
solve geometry rider)
As an interrelationship
of different systems.
Circle in 2-dimension is
extended to include a
sphere in three-
dimensional space.
2.1.4 How does the model work
The different levels of the model do not function independently of each other. The
different levels are closely linked to a “network of relations” (van Hiele, 1973 cited in
Human et al, 1979:20). Human et al (1979) quote van Hiele (1973) who describes this
network of relations between the different levels as: “In a network of relations the
words ‘rhombus’, ‘side’, ‘square’, etc, have unique meanings with a distinct
collection of properties. Each level is associated with a different network of relations”
(Human et al, 1979:20).
It is important for educators to be aware of how the van Hiele model works, since
knowledge thereof would impact on the instructional strategies to be used. Below is a
brief sketch, according to Crowley (1987), as to how the van Hiele model works.
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Sequential: For a learner to function adequately (or competently) at level 1 for
instance, the learner should have grasped the basics of the previous level adequately.
This type of pre-requisite building blocks is akin to the Piagetian concepts of
“assimilation and accommodation” (Helms and Turner, 1981:51). Thus like the van
Hiele model which emphasizes an orderly growth path, Piaget’s theory also premises
all “intellectual behaviour has its beginnings in early infancy, and mature reasoning
skills emerge through subsequent phases of conceptual development” (Helms and
Turner, 1981:51).
Advancement: The learner’s progression from one level to the next is more
dependent on the instruction received than the biological maturity of the learner. It
should be noted that no method of instruction allows learners to skip levels, i.e. a
learner cannot move from level 0 to level 2, without first experiencing level 1.
However, methods of instruction (teaching) can “enhance progress, whereas others
retard or even prevent movement between levels” (Crowley, 1987:4).
Intrinsic and extrinsic: Initially learners are able to recognize figures and shapes on
the basis of their physical appearance. This phase on focusing on some part of the
figure/shape is similar to Piaget’s concept of centering (Helms and Turner, 1981). The
concept of centering is when a learner develops a tendency to “concentrate on a single
outstanding characteristic of an object while excluding its other features” (Helms &
Turner, 1981:41). For instance a grade 4 learner is shown shapes of squares and told
that those are squares. It is not until later in the learner’s life that “the figure is
analysed and its components and properties are discovered” (Crowley, 1987:4).
Linguistics: Each level of the model has its own set of terminology, which is
appropriate for the learner at that particular stage. Crowley (1987) cites van Hiele
(1984), in which the latter asserts, “Each level has its own linguistic symbols and its
own system of relations connecting them” (van Hiele, 1984, cited in Crowley,
1987:4). For instance learners in lower grades, when taught multiplication are often
told that when multiplying two numbers the answer is always more or equal to the
two numbers, e.g.: 3 X 4 = 12, 1X 2 = 2, etc. However, as these learners progress in
their schooling careers and are exposed to rational numbers then that particular rule is
- 29 -
no longer valid, since 2 X ½ = 1, which is less than 2 and more than ½. In a like
manner in geometry then, “a figure may have more than one name (class inclusion) - a
square is also a rectangle (and a parallelogram)” (Crowley, 1987:4). But for a learner
at the van Hiele level 1, this does not make sense. “This type of notion and its
accompanying language, however, are fundamental at level 2” (Crowley, 1987:4).
Mismatch: De Villiers (1997) captures the high failure rate in Euclidean Geometry
aptly when he states, “the curriculum was presented at a higher level than those of the
pupils; in other words they could not understand the teacher nor could the teacher
understand why they could not understand” (p.41). If the educator (teacher), teaching
materials, subject matter, language, etc. do not cohere with the learners level of
development, then the end result will be a learner lacking the ability to “follow the
thought processes being used” (Crowley, 1987:4).
2.1.5 Learning Phases
As has been alluded to above, movement through the levels of the van Hiele model is
dependent more on the type of instruction received than on the age level or maturation
of the learners. “Thus the method and organization of instruction, as well as the
content and materials used, are important areas of pedagogical concern” (Crowley,
1987:5). To address the issues around content and instructional tools to be used the
van Hiele s had identified five areas (phases) of learning that will assist the educator.
The five areas are “inquiry, directed orientation, explication, free orientation, and
integration” (Crowley, 1987:5). According to the van Hieles, if a topic or section of
geometry is taught according to the above sequence, learners’ will be able to gain
mastery of a particular level. The above phases of learning/ teaching are present at
each level of the van Hiele model.
Each of the five phases of learning is explained below.
Phase 1: Inquiry / information: At this primary stage learners and educators are engaged in conversation with each
other about the topic at hand. Learners make observations related to the task, ask
clarity seeking questions and the educator should introduce vocabulary pertinent to
- 30 -
the specific level at which the task is dealt with. For instance, the educator may ask
learners to distinguish between a cyclic quadrilateral and any convex quadrilateral. Is
a parallelogram a cyclic quadrilateral? Is a rectangle a cyclic quadrilateral? When
will a parallelogram be a cyclic quadrilateral? Why do you say that?
Why should a teacher be engaged in such activities? Teachers’ need to engage in such
an activity as it serves a dual purpose, viz:
(i) “The teacher learns what prior knowledge the students have about a topic;
and
(ii) The students learn what direction further study will take” (Crowley,
1987:5).
Phase 2: Directed orientation
At this phase, the learners’ begin to explore a topic using material that has been
carefully sequenced by the educator. The activities should reveal to the learners the
features peculiar to that particular level “Thus, much of the material will be short
tasks designed to elicit specific responses” (Crowley, 1987:5).
Phase 3: Explication
Building on their previous experiences learners begin to express and share their views
about the observations made regarding a concept. During this stage the educator plays
a minimal role. The educator’s role is restricted to assisting learners acquiring and
using “accurate and appropriate language” (Crowley, 1987:5). It is during this phase
that particular levels of systemic relations begin to become apparent.
Phase 4: Free orientation
During this period of learning, learners are exposed to and engaged in open –ended
tasks that can be completed in a variety of ways. The tasks are non-routine, multi-
stepped, complex tasks. Consider the following example: In the given diagram, below
- 31 -
left, BD is a diameter of the circle. ABCD is a cyclic quadrilateral. BC = 5 cm, AC =
6 cm and BD = 8 cm. Calculate the length of AB.
Figure 4: An example of how learners’ knowledge from different section of the Mathematics can be recruited to solve a given problem.
Phase 5: Integration At this stage of learning, learners need to bring together (synthesize), what they have
learnt, with the aim of forming an overview of the new relationships of objects and
relations. The educator can assist by “furnishing global surveys” (van Hiele, cited in
Crowley, 1987:6). At this stage rules may be formulated and memorized for use later.
By the end of phase five (integration), learners have attained a new level of thinking.
This new level of thinking replaces the previous level of thinking and learners are
once again ready to repeat the five phases of learning at the next level of the van Hiele
model of thinking.
B
D
C
A
Begin by calculating the length of DC in ∆BDC using
the Theorem of Pythagoras, i.e. DC = 39 .
Next, calculate the size of CDB∧
using the
trigonometric ratio of tan, i.e. tan CDB∧
=39
5
CDB∧
= 38,70 = CAB∧
(angles in same segment). Using the above results, we proceed to calculate AB, using the cosine rule.
In ∆ABC: (BC)2 = (AB)2+(AC)2 –2(AB)(AC)cos CAB∧
Through substitution and simplification we arrive at the following values for AB: AB = 8,0 or AB = 1,4
- 32 -
Table 2: The van Hiele model of thinking together with the phases of learning Van Hiele Level Phases of learning
Level 4 (Rigour) Integration
Free orientation
Explication
Directed orientation
Level 3 (Deduction) Integration
Free orientation
Explication
Directed orientation
Level 2 (Informal deduction) Integration
Free orientation
Explication
Directed orientation
Level 1 (Analysis) Integration
Free orientation
Explication
Directed orientation
Level 0 (Visual) Integration
Free orientation
Explication
Directed orientation
2.1.6 The role of language
“Effective learning occurs as students actively experience the objects of study in
appropriate contexts of geometric thinking and as they engage in discussion and
reflection using the language of the period” (Teppo, 1991:213). Language is a key
component of learning. According to the Department of Education (DoE), one of the
aims of mathematics is for learners to “develop the ability to understand, interpret,
read, speak and write mathematical language” (1995). The role of language in
geometry cannot be understated. Language or language appropriate to the learner’s
level of thinking, as well as the identification of suitable material, are pivotal aspects
in the development of the learners’ geometric thinking.
According to van Hiele, the primary reason for the failure of the traditional geometry
curriculum can be attributed to the communication gaps between teacher and learner.
De Villiers (1997) captures it aptly when he writes, “they could not understand the
teacher nor could the teacher understand why they could not understand!” (p.41). In
- 33 -
order to enhance conceptual understanding it is important for learners to communicate
(articulate) their “linguistic associations for words and symbols and that they use that
vocabulary” (Crowley, 1987:13). Verbalizations call for the learners’ to make a
conscious effort to express what may be considered vague and incoherent ideas.
Verbalisation can also serve as a tool to expose learners “immature and misconceived
ideas” (Crowley, 1987:14). At first learners should be encouraged to express their
geometric thinking in their own words, e.g.: “Z-angles” for alternate angles; a
rectangle that has been kicked” for a parallelogram, etc. As learners advance in their
geometry studies at school they should be exposed to the appropriate terminology and
encouraged to use it correctly.
A learner’s usage of a word (or term), in mathematics does not imply that the teacher
and the listener (the learner) share the same meaning of the word used. For instance
when a teacher uses the word parm (short for parallelogram) is the listener thinking of
a parallelogram or the palm of his / her hand? As another example, if a learner is
given a square in standard position, i.e. � , the learner is able to identify the figure as
a square, but if its rotated 450 like , then it’s no longer a square. In the example
the learners’ focused on the orientation of the figure as the determining fact of the
“squarness” of the figure. By engaging learners’ in discussion and conversation,
educators can expose learners’ misconceptions and incomplete ideas as well as build
on correct perceptions.
For the learner to acquire and correctly utilize the appropriate language, the role of the
educator becomes paramount. For example, if the learners’ are working at level 1 of
the van Hiele model, then the educator should be seen to be using terms such as “ all,
some, always, never, sometimes” (Crowley, 1987 : 14). As the learner progress along
the van Hiele continuum, appropriate terms need to be used. Terms or phrases, which
are typical for some of the levels, are:
Table 3: Terms used at levels 2 and 3 of the van Hiele model
Level Terms / Phrases
2 If [condition] then [results] it follows that
3 Axioms, postulate, converse, necessary and sufficient; theorem etc.
- 34 -
The type of questions posed by educators’ is the key in directing learners’ thinking.
Questions that require regurgitation of information supplied by the educator will not
foster critical thinking, which is needed for geometry. Learners’ need to be able to
explain and justify their explanation in a critical manner, “Mathematics is based on
observing patterns; with rigorous logical thinking, this leads to theories of abstract
relations” (DoE, 2003:9). Learners’ should be challenged to explain “why” as well as
to think about alternative approaches to their initial explanation, by posing appropriate
questions, allowing sufficient waiting time and engaging learners’ in discussion of
their answers and methods which consider learners’ level of thinking.
For growth in the learners thinking to happen, the level of instruction of the teacher
needs to correspond with the learners’ level of development. Thus the educator must
be able to ascertain the learner’s level of geometric thought, for each of the levels in
the van Hiele model are characterized by their own unique vocabulary which is used
to identify the concepts, structures and networks at play within a specific level of
geometric thinking. “Language is useful, because by the mention of a word parts of a
structure can be called up” (van Hiele 1986, cited in Teppo, 1991 : 231) .
2.1.7 Conclusion
Van Hiele’s model of geometric thought, as well as their phases of learning is a
constructive attempt to assist in identifying a learner’s stage of geometric thought as
well as the means to progress through the levels. Progression through the levels is
dependent more on the type of instruction received than on the learners’ physical or
biological maturation level. The model has been used extensively in different research
studies (e.g. : Burger 1985 ; Burger & Shaunghnessy, 1986) to assess learners’
understanding of geometry. The model if applied appropriately throughout the
schooling phase [i.e. grade 1 -12], will result in geometric thinking becoming
accessible to all.
- 35 -
2.2. Literature Review
2.2.1 Introduction
A snap survey of the writings on school geometry tends to point to two main
problems. Firstly, poor learner performance in the subject, e.g. the De Villiers and
Njisane’s 1987 study of grade 12 learners in KwaZulu Natal whose performance they
note that “ 45% of black pupils in grade 12 (Std. 10) in Kwazulu had only mastered
level 2 or lower, whereas the examination assumed mastering of level 3 and beyond”
(De Villiers , 1997 : 42). The poor performance of secondary school learners’ in
Geometry has also been corroborated by other studies such as Malan (1986); Smith &
De Villiers(1990) and Govender(1995).
A second contributing factor to a learner’s poor performance in Euclidean geometry is
the nature of the curriculum currently in use. I have alluded earlier in the discussion
that “South Africa is the habitat of an endangered species, for Euclidean geometry has
disappeared from the syllabus of most other countries” (Mathematical Digest, July
1996, as citied in De Villiers , 1997 : 37 ). While South Africa may be the last
surviving bastion of a “not so popular” branch of Mathematics, it is worthy to note
that in recent times geometry at all levels has undergone a rebirth or revival in most
countries (De Villiers, 1997). It should be noted that the concern about an archaic
curriculum is nothing new – as early as 1969, Allendoefer commented on the
American curriculum as follows:
“The mathematical curriculum in our elementary and secondary school faces a serious dilemma when it
comes to geometry. It is easy to find fault with the traditional course in geometry, but sound advice on
how to remedy the difficulties is hard to come by……Curricular reform groups at home and abroad
have tackled the problem, but with singular lack of success or agreement……We are therefore, under
pressure to “ do something” about geometry ; but what shall we do ? (Usiskin, 1987: 17)
Each of the problems identified in the preceeding paragraphs will now be explored in
some detail.
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2.2.2 The problem about Curriculum
The current grade 10-12 Higher Grade and Standard Grade syllabus for mathematics
strives to foster amongst others the following learning and teaching aims:
2.2.2. critical and reflective reasoning ability;
2.2.4. fluency in communicative and linguistic skills e.g. reading, writing,
listening and speaking;
2.2.8. to contextualise the teaching and learning in a manner which fits the
experience of the pupils” (DoE, 1995)
In addition to the above learning and teaching aims, the same syllabus documents list
the following aims peculiar to mathematics, which need to be fostered and developed
in learners.
2.3.1. to enable pupils to gain mathematical knowledge and proficiency ;
2.3.3. to develop insight into spatial relationships and measurements ;
2.3.4 to enable pupils to discover mathematical concepts and patterns by
experimentation, discovery and conjecture ;
2.3.6. to develop the ability to reason logically, to generalize, socialise,
organize, draw analogies and prove;” ( DoE , 1995 )
Both sets of aims in the syllabus documents are designed to ensure that learners are
actively engaged in the “construction” of knowledge. The content should correlate
with the learner’s experiences (see 2.2.8 above) and offer learners appropriate
opportunities to develop the “geometric eye” (see 2.3.3 above) through a process of
discovery, experimentation, hypothesizing and so forth. However, when one peruses
through a grade 12 textbook, one observes that textbook authors have failed learners
by not providing them with sufficient experiences to travel the path of the van Hiele
model. Often, textbook authors provide learners with a finished product of the proof
of a theorem only.
- 37 -
e.g. 1: “Theorem 1:
A line parallel to one side of a triangle cuts the other two sides, or these sides
produced, proportionally.”
(Bopape, Hlomuka, Magadla, Shongwe, Taylor, Tshongwe, 1994:186)
Figure 5: A typical proof which grade 12 learners are expected to reproduce under test / examination conditions.
Given: ∆ ABC, with X on AB and Y on AC. XY // BC
RTP:
YCAY
XBAX =
Construction: Draw XC and YB. Draw altitudes XK ⊥ AC (or AC produced) and YH ⊥ AB (or
AB produced)
Proof: Area ∆ AXY = AX.YH
2
1
Area ∆ BXY = BX.YH
2
1
∴
XBAX
BXY∆areaAXY∆area =
Similarly
YCAY
CXY∆areaAXY∆area =
But Area ∆ BXY = Area ∆ CXY (same base, same height)
∴
YCAY
XBAX =
H K
Y
A
BC
X
- 38 -
e.g. 2: “Theorem 5(a) :
The opposite angles of a cyclic quadrilateral are supplementary (Opp. <’s of cyclic
quad.)” (Laridon, Brink, Fynn, Jawurek, Kito, Myburg, Pike, Rhodes- Houghton,
Van Rooyen, 1987:319)
Figure 6: A typical proof which is presented to Grade 11 learners’ as a finished product.
Given: Circle O containing cyclic quad. ABCD
Required to prove: 0180=+
∧∧CA
and 0180=+∧∧DB
Proof: Draw BO and DO
∧∧= A2O1 (< at centre = 2 X < at circle)
∧∧= C2O2 (< at centre = 2 X < at circle)
∴
+=+
∧∧∧∧CAOO 21 2
But 0360=+
∧∧
21 OO
∴ 0180=+
∧∧CA
Similarly, by joining AO and CO, we could prove 0180=+
∧∧DB .
2
1O
DB
A
C
- 39 -
Both examples, sourced from different textbooks demonstrate how learners are given
the theorems as a finished product. When I speak of a product in mathematics, it is
meant here “the end-result of some mathematical activity which preceded it” (De
Villiers, 1997: 45).In the examples cited above the learners were not (if teachers
follow textbooks slavishly) afforded the opportunities to develop the skills articulated
under 2.3.4 above. The mathematical processes are not allowed to be developed fully
within learners - although official policy documents, such as the syllabus encourages
such processes to be nurtured and developed within learners. The process vs. product
dichotomy is not new to our schools.
As early as 1978, the predecessor to Association of Mathematics Education of South
Africa (AMESA), the Mathematical Association of South Africa (MASA) noted
regarding changes to the math’s syllabus in South Africa that :
“The intrinsic value of mathematics is not only contained in the PRODUCTS of
mathematical activity (i.e. polished concepts, definitions, structures, and axiomatic
systems), but also, and especially, in the PROCESSES OF MATHEMATICAL
ACTIVITY leading to such products, e.g. generalization, recognition of pattern,
defining, axiomatising. The draft syllabi are intended to reflect and increase emphasis
on genuine mathematical activity as opposed to the mere assimilation of the finished
products of such activity. This emphasis is particularly reflected in the various
sections on geometry” (MASA, 1978, cited in De Villiers, 1997: 45).
The good intentions cited above as well as those cited in the 1995 syllabus seem to
have fallen on deaf ears. Teachers and many textbook authors continue providing
learners with ready made content, especially in geometry, which learners had to then
“assimilate and regurgitate in tests and exams” (De Villiers, 1997 : 45) thereby
confirming the assertion that geometry is useless outside the classroom.
By perpetuating the current product-driven approach to geometry, it will only increase
learners’ negative attitude to the discipline. Sentiments such as the following
“but I don’t see how geometry will be of any value to me ……… We have to get all the
statements in just the right order on the left side of the page and always write some reason
for the statement on the right-hand side of the page. We memorize definition after
definition for things we already know.” (Joubert, 1988: 7),
- 40 -
does little to light up the discipline of geometry. Our dogmatic approach to two-
column proofs and learners’ aimless memorization of definitions of objects they
already know add salt to the learners’ wounds when dealing with geometry. De
Villiers (1998) argues from a theoretical vantage point that instead of teaching
learners definitions of geometric concepts such as quadrilaterals, we should rather
strive to develop students’ ability to define.
Mathematicians such as (Blandford, 1908 and Freudenthal, 1973, as cited in De
Villiers, 1998) are strong proponents allowing learners’ to be actively engaged in
coming up with definitions for geometric concepts. Blandford (1908), in De Villiers
(1998) regards the method of giving learners’ ready made definitions as a “radically
vicious method” (in De Villiers, 1997: 46), and by so doing one is robbing the
learners’ of the most intellectually enriching activities. “The evolving of the workable
definition by the child’s own activity stimulated by appropriate questions is both
interesting and highly educational.” (De Villiers, 1997: 46).
Researchers like Ohtani (1996, in De Villiers, 1998) have argued that the traditional
method of providing learners with ready-made definitions by the teacher is an attempt
by the teacher to exercise his control over learners, to avoid any dissension, and not
having to deal with students’ ideas as well as to steer clear of any “hazardous”
interaction with learners. The student’s ability to regurgitate a definition of a cyclic
quadrilateral does not imply that the learner understands the concept (Vinner, 1991, in
De Villiers, 1998). For example, a learner may be able to recite the standard definition
of a cyclic quadrilateral as “A cyclic quadrilateral is a quadrilateral of which the
vertices lie on a circle” (Laridon, et al, 1995: 277), but the learner may not consider
that if a quadrilateral with exterior angle equal to the interior opposite angle as being
cyclic, since the learners’ concept map of cyclic quadrilaterals does not include cases
where points are not on a circle. “I would appeal that in order to increase students’
understanding of geometric definitions, and of the concepts to which they relate, it is
essential to engage them at some stage in the process of defining geometric concepts”
(De Villiers, 1998 : 2).
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2.2.3 Learners’ views of proof for a geometrical p roblem
There has been a growing interest in mathematics education in recent times regarding
the teaching and learning of proof (e.g. Hanna, 2000; Dreyfuss, 1999; De Villiers,
1997; 1990). Whilst there has been this resurgence in proof across mathematics fields
at school and tertiary level, there has also been a lack of interest in the mathematical
reasoning which learners’ engage in when solving geometrical riders.
Traditionally proof has been seen primarily as a means to verify the accuracy
(correctness) of mathematical statements (De Villiers, 1990). However, this
stereotypical, constrained view of proof has been criticized in recent times by,
amongst others, De Villiers (1990), Hanna (2000), and Dreyfuss (1999). De Villiers
(1990), for instance argues that proof in mathematics is more than just for verification
purposes. He maintains that the view held by most people in mathematics education
that verification is the cornerstone of proof is avoiding “the real nature of proof”
(Bell, 1976, in De Villiers, 1990:18), as verification in mathematics can be obtained
using “quite other means than that of following a logical proof” (De Villiers,
1990:18).
For De Villiers (1990), and others (like Hanna, 2000), proof is made up of the
following processes:
• Verification (concerned with the truth of a statement);
• Explanation (providing insight into why it is true);
• Systematization (the organization of various results into a deductive system
of axioms, major concepts and theorems);
• Discovery (the discovery or invention of new results) ;
• Communication (the transmission of mathematical knowledge) (De Villiers,
1990:18).
Whilst the above five aspects will not be dealt with in any detail here, it is sufficient
to state that proof is akin to van Hiele’s level 3 stage of reasoning. In the van Hiele
model of geometric reasoning, level epitomizes the learners’ ability to understand “the
interrelationship and role of undefined terms, axioms, postulates, definitions,
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theorems, and proof is seen” (Crowley, 1987:3). Another characteristic of the learner
at this level of geometric thinking is the learner’s ability to “construct, not just
memorize proofs” (Crowley, 1987:3).
The tasks given to the learners in this study were designed not to test their ability to
regurgitate theorems, but to check their understanding of the theorems, which they
had encountered at school level. Unlike other branches of school mathematics, which
are largely algorithmic in nature, geometry is different. The solution to a geometrical
rider is in essence a learner’s explanation, using theorems, axioms and properties of
the figures involved. The thrust of this research project is not on proof per se, but the
manner in which learners’ present their solutions, which is indicative of the learners’
understanding of the nature and purpose of proof.
Like their fellow high school colleagues in other countries, these students’ concept of
proof corresponds with their international counterparts: “most high school and college
students don’t know what a proof is or what it is supposed to achieve” (Dreyfuss,
1999:94). At high school level, the distinction between a proof, an explanation, and an
argument is not always clear. Whilst this is not the focus of this study, I’d like to end
with Hanna’s (1995) observation that “while in mathematical practice the main
function of proof is justification and verification, its main function in mathematics
education is surely that of explanation” (p. 47).
2.2.4 The Performance problem
When compared to other branches of mathematics (e.g.: Calculus, Trigonometry, etc),
learners’ performance in Euclidean Geometry is dismal. Comments such as
• “This [Euclidean Geometry] is probably still the least well-done of all
sections” (GDE, 2003 : 145 ),
• “This [Euclidean Geometry] of the work… is still not well done on the whole.
Many candidates write down geometric information but it often does not make
sense nor is it relevant to a particular question. They don’t appear to
understand the words they are using.” (GDE, 2002 : 102),
are indicative of learners’ poor performance in Euclidean Geometry.
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The following tables, which have been extracted from the former House of Delegates
examiners’ report, shed clearer light on learners’ performance on both HG and SG,
grade 12 paper 2 sections. Both HG and SG learners at grade 12 level are tested on the
following aspects: Trigonometry, Analytic Geometry and Euclidean Geometry.
Table 4 below shows the Higher Grade learners’ performance in the different sections
of the syllabus relevant to paper 2, written in 1991.
Table 4: Performance of Higher Grade grade 12 learners in Paper 2 (ex HOD, 1993: 1)
From the above table one can infer that the bulk of the learners’ (68%) scripts
sampled in 1991, scored between 0% - 39% in Euclidean Geometry. Furthermore, on
average, learners’ scored 32% for Euclidean Geometry, which translates to a raw
mark of 22, 4 out of 70. When compared to Trigonometry, learners’ scored on
average 65.6/80 (82%) and in Analytic geometry learners scored on average 31/50
(62%) – then the 22, 4/70 is indeed poor – especially for learners on the higher grade.
Table 5, below is an indication of how Standard Grade, grade 12 learners performed
in the 1991 examinations.
Table 5: Standard Grade learners average percentage in different sections of Paper 2 (ex HOD, 1993: 1)
NOTE:
1. The data in both Tables 4 and 5 above is based on a 10% sample of
learners’ who wrote the 1991 grade 12 exams.
1 Euclidean Geometry is also at times referred to as Synthetic Geometry
Performance of Candidates
TOTAL
MARKS
50% -
100%
40% -
49%
0% -
39%
AV %
of PASS
Trigonometry 80 60% 22% 18% 82%
Synthetic Geometry1 70 22% 10% 68% 32%
Analytic Geometry 50 36% 26% 38% 62%
SECTION TRIGONOMETRY SYNTHETIC
GEOMETRY
ANALYTIC
GEOMETRY
AVERAGE % 59 32 26,5
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2. Prior to 1996, the South African education spectrum was divided into
almost 20 different entities. Each entity catered for the specific
grouping of the population.
3. The data in Tables 4 and 5 is based on the former House of Delegates
report, which catered for the Indian population of the South African
community. I have used these results because I had access to them
since my first year of teaching in 1992.
From the data in Tables 4 and 5 above we notice that amongst “Indian” learners
sitting for the grade 12 exams, Euclidean Geometry was the component in the second
paper where both higher grade and standard grade learners faired poorly.
More recently, the Mathematics, Science and Technology (MST) project team in
Gauteng was commissioned by the Gauteng Department of Education (GDE) to
provide an analysis of learners’ performance in Mathematics, Science and Biology at
grade 12 level. The MST team used a sample of 2002 Gauteng grade 12 learners for
analysis purposes. The team “perused through 30 standard grade and 31 higher grade
scripts to get a superficial sense of how the candidates went through the questions”
(MST Report, 2003: 3).
Table 6 provides us with an indication of standard grade learner’s performance in
some of the questions in the second paper.
Table 6: Standard Grade learners performance in some questions of the question paper (Maths Standard Grade Paper 2) (MST, 2003:6)
NO LEARNERS WHO SCORED
BETWEEN
QUESTION
MARKS
No. of learners not attempting a
question
0 – 39%
40 – 59%
60 – 100%
TOTAL
1 18 4 6 3 17 30
3 20 5 4 5 16 30
4 12 6 8 8 8 30
6 13 13 8 5 4 30
8 18 8 12 1 9 30
9 19 7 14 2 7 30
TOTAL 100
Questions 8 and 9, in the above table, are questions involving Euclidean Geometry.
We note that question 8 involved a rider dealing with the tangent-chord theorem and
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cyclic quadrilaterals. From the cohort of 30 learners, 20 of the 30 or 67% of the
candidates are unable to deal with riders involving cyclic quadrilaterals and tangent-
chord theorems. Question 9 dealt with similar triangles and in this question 70%
[21/30] learners are unable to either begin to solve the problem or have attained
between 0 and 7 out of a possible 19 marks.
Table 7 below provides us with a snapshot of 31 higher-grade learners’ performance
in a sample of questions in the 2002 exams. Whilst other questions are reflected as
well, our focus is on Euclidean Geometry.
Table 7: HG learners’ performance in Mathematics 2002 Grade 12 question paper. NO LEARNERS WHO SCORED BETWEEN
Question
Marks
Not attempted by learners 0 – 39%
40 – 59%
60 – 100%
TOTAL
1 23 10 10 7 10 31
2 25 7 7 1 16 31
4 22 5 10 6 10 31
5 19 4 11 6 10 31
6 24 7 13 2 9 31
7 19 13 13 1 4 31
8 25 10 14 2 5 31
9 12 21 0 4 6 31
TOTAL 169
In the above table (Table 7) we note that 26 / 31 higher-grade learners’ are unable to
handle geometric riders involving cyclic quadrilaterals (Question 8). Questions 9 and
10 dealt with similar triangles, which are also a cause of concern – but it is not the
focus of this study.
We notice that whilst table 4 and 5 may refer to “Indian” learners in the main, Tables
6 and 7 are “race blind” in that the samples of scripts selected would include
candidates from across the racial divide. Thus one may have been tempted to regard
Euclidean Geometry as problematic only amongst Indian learners, but Tables 6 and 7
suggests that in South Africa, Euclidean Geometry is a problem endemic to all
schools and communities within South Africa.
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Learners’ poor performance in Geometry, especially with regard to proofs of
theorems, can be attributed to learners’ memorization of proofs of theorems. A typical
comment by examiners is that “ Pupils are swotting off Theorems and are thus unable
to provide a logical proof of what is provided” (GDE, 2000: 3). In terms of the van
Hiele model of thinking the ability to construct and understand proofs is located
within level 3. However, because learners have not gained mastery at the lower levels
of the model (i.e. Level 2, 1, 0); they resort to rote learning of the proofs of theorems.
Furthermore when learners are asked to prove that a quadrilateral is cyclic and no
circle is present learners experience difficulty in accomplishing such tasks (GDE,
2000).
Although, I have been alluding to learners’ poor performances at grade 12 levels in
Euclidean Geometry, for improvement and a change of attitude towards secondary
school geometry, the primary school geometry curriculum needs to be redesigned to
be aligned with the levels of geometric thought according to the van Hiele model.
“The future of secondary school geometry thus automatically depends on primary
school geometry!”(De Villiers, 1997: 43).
A learner’s ability to provide meaningful proofs for geometric theorems and riders
epitomizes the learner’s development in geometry, according to the van Hiele model
of thinking. However, in reality this is not the case. Examiners’ reports (GDE, 2001,
2002, MST, 2003, TED, 1994, HOD, 1993) all lament learners’ poor performance
when asked to prove theorems or solve riders. Learners’ ability or in-ability to provide
successful proofs is best explained in terms of Piaget’s theory and van Hiele’s model
of geometric understanding. Both van Hiele’s model and Piaget’s theory suggest that
learners must progress through the lower levels of geometric thinking before they can
gain mastery at higher levels such as the writing and understanding formal proofs.
The route travelled from the lower levels to the higher levels of thinking takes a
considerable amount of time. “The van Hiele theory suggests that instruction should
help students gradually progress through lower levels of geometric thought before
they begin a proof – orientated study of geometry” (Battista and Clements, 1995: 50)
Very often, educators bypass the different levels of the van Hiele model and hope that
learners will understand what the teacher has taught. This type of naïve, premature,
dealing of formal proof results in learners resorting to memorization and “confusion
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about the purpose of proof” (Battista & Clements, 1995: 50) sets in. Both the van
Hiele model and Piaget’s theories suggest that learners’ can understand and work with
an axiomatic deductive system only once they have gained mastery in the highest
levels of both theories. “Thus, the explicit study of axiomatic systems is unlikely to be
productive for the vast majority of students in high school geometry” (Battista &
Clements, 1995:50).
The studies undertaken by other researchers, using the van Hiele model of geometric
thought restricted their investigations to linear figures such as squares and rectangles.
However, not much research has been undertaken that involves the use of the van
Hiele model to ascertain learners’ reasoning ability involving shapes other than
rectangles and squares and at a level such as grade 12. This study contributes to the
existing knowledge base of the Van Hiele levels of understanding. The study deals
with the following research questions:
1. How do grade 12 learners begin to solve a geometric problem?
2. What knowledge and skills do learners recruit in order to prove geometric
problems?
3. How do learners justify that a proof to a geometric problem is complete?
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Chapter 3: Research design and methodology
3.1. The Research Design
There exist three common research designs, which dominate the educational
landscape. Reeves and Hedburg (2001) have identified them as the quantitative,
qualitative and the eclectic-mixed mode pragmatic research model. Researchers
showing preference to work in the quantitative paradigm present their results
primarily in the language of numbers. In a quantitative paradigm, the purpose is “not
to report data verbally, but to represent those data in commercial values” (Leedy
1993: 243). One just needs to caution the reader here and state that it does not mean
that the other paradigms do not use calculations when doing analysis of data.
However, calculations are “not the major form in which the data exist” (Leedy, 1993:
243). Data collected in a quantitative paradigm is usually analysed using inferential
and descriptive statistical techniques.
This study aims to be exploratory and interpretative in nature, thus data collection and
analysis thereof will be primarily determined in relation to the contextual setting and
perspectives of the learners. Researchers working within a quantitative paradigm are
keen on giving a “more holistic picture of what goes on in a particular situation or
setting” (Fraenkel & Wallen, 1993: 10), i.e. they (quantitative researchers) are more
interested in gaining a richer understanding of the social phenomena at play than the
quantitative data only.
Silverman (2000) cites Hammersly (1992) in which the latter identified five
preferences of quantitative researchers. The five preferences are listed in the Table 8,
below.
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Table 8: The preferences of qualitative researchers (Silverman, 2000:8)
From Table 8, above one should not conclude that practices such as hypothesis testing
do not feature as part of the qualitative paradigm. McMillan and Schumacher (2001)
argue that in qualitative paradigm researchers’ use strategies which are flexible,
“using various combinations of techniques to obtain valid data” (p.396). This
flexibility of utilization of approaches is encapsulated in the eclectic-mixed mode of
research, which characterizes cooperation between the qualitative and quantitative
paradigms when collecting data related to educational problems (Reeves and
Hedburg, 2001). As alluded to earlier, this study is inquiry- based and interpretive,
leaning more towards a qualitative research paradigm than it does with the other
paradigms. This does not imply that the study was conducted exclusively without due
regard to the quantitative and or eclectic-mixed methods to highlight points of interest
that arose in the study, which could be explored in more extensive and further
research analysis.
The overarching research approach for this study was a case study design.
“Qualitative research uses a case study design meaning that the data analysis focuses
on one phenomenon, which the researcher selects to understand in depth regardless of
the number of sites or participants for the study” (McMillan and Schumacher,
2001:398). The interpretive nature of case studies allows the researcher to “study and
give insight into specific situations or events” (Stake, 1995). In this study the
phenomenon explored is grade 12 learners’ understanding of Euclidean Geometry. In
case studies, generalizations of findings within a wider population or community are
not of paramount importance. Fraenkel and Wallen (1993) maintain that a great deal
1 A preference for qualitative data understood simply as the analysis of words and
images rather than numbers.
2 A preference for naturally occurring data- observation rather than experiment,
unstructured rather than structured interviews.
3 A preference for meanings rather than behaviour- attempting “to document the
world from the point of view of the people studied” (Hammersly, 1992).
4 A rejection of natural science as a model.
5 A preference for inductive, hypothesis-generating research rather than hypothesis
testing (cf. Glasser and Strauss, 1967)
- 50 -
can be “learned from studying just one individual, one classroom, or one school
district” (p.392). Elsewhere, Silverman (2000) argues that in qualitative research, the
focus should not be on generalizations. Silverman (2000), cites Alasuutari (1995),
who notes that
“Generalization is… [a] word. that should be reserved for surveys only. What can be analysed
instead is how the researcher demonstrates that the analysis relates to things beyond the
material at hand…extrapolation better captures the typical procedure in qualitative
research” (Alasuutari, 1995, in Silverman, 2000:111).
A single case study design was an appropriate research tool in this study as it afforded
the researcher the opportunity to explore strategies learners’ used in solving geometric
riders, discover important questions to ask relative to learners’ reasoning strategies
employed when solving geometric problems. According to Bell (1987), the great
advantage of the case study lies in the fact that it “allows the researcher to concentrate
on a specific instance or situation and to identify, or attempt to identify, the various
interactive processes at work” (p.6), which may not be easy to identify in a large-scale
survey study.
The above discussions on case study design indicate their usefulness as an appropriate
and useful method for investigating processes in education, and were thus employed
for this study.
3.2 Access to participants
The principal participants in this study were one educator and a cohort of her grade 12
learners from a co-educational school in the Tshwane South District, in Gauteng.
Anecdotal evidence, such as discussions with educators around geometry always end
or begin with “My learners hate geometry. I have tried everything but nothing seems
to help. I will just teach the way I have been doing –theorem → example → past
question papers and that’s it! (emphasis added). Despondency, as manifested above,
comes from an educator who has been voted as Gauteng’s Teacher (Mathematics /
Science) of the year in 2004 and 2003. However, what she did not succeed to do was
to improve her learners’ performance in Geometry. Her learners scored well in other
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sections of maths but performed poorly in Geometry as a result their final marks were
being affected.
The selection of the teacher was based on the interactions I had with her during grade
12 marking sessions, workshops and on-site school visits. No teacher is an island and
schools displaying positive leadership structures and allowed their teachers to make
effective and innovative use of resources were also considered. Furthermore
feasibility [distance and transport costs, etc] for the duration of the study was also a
factor to be considered. In conclusion and perhaps most importantly, the willingness
and cooperation of the educator, the learners and the school management team for the
duration of the study (six months) was necessary for the case study to be of significant
value.
My first interaction with the educator occurred in January 2004, via telephone to set
up an informal meeting with the educator and principal. The meeting was scheduled
for early February 2004, and took place at the school of the identified educator. At
this meeting the educator and principal were briefed about the purpose of the study,
the reason(s) for considering the teacher to be an appropriate participant in the study,
the data collection tools to be used, the duration of the study and the commitment and
willingness of the educator to ensure that the study is meaningful. At this preliminary
meeting, the educator (who will henceforth be referred to as Teacher) agreed to
participate in the study and was eager to share with me information regarding her
learners’ performance in Euclidean Geometry. The Teacher raised no objection to the
data collection methods to be used, provided prior approval was granted by the school
management team to record part or some of the lessons delivered and consent was
obtained from the learners’ to be interviewed.
Subsequent to our initial meeting, a formal letter (see Appendix 1) was submitted to
the Principal, in which permission was sought to conduct research at the school with
the identified learners. The letter also explained the nature and expected duration of
the study and how at the end, the learners and teachers of mathematics could benefit.
Permission from the Gauteng Department of Education (GDE) was also sought to
conduct research by informing the relevant units (e.g. Policy and Planning Unit) about
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the nature and duration of the study. Data collection was tentatively planned to be
conducted during August and September 2004.
3.2.1 Data collection instruments
The triangulation of data is used in qualitative research as a means to seek patterns in
the data collected (McMillan & Schumacher, 2001). According to McMillan &
Schumacher (2001), the triangulation of data in its broadest sense is not restricted to
data only. It can include “use of multiple researchers, multiple theories, or
perspectives to interpret the data; multiple data sources to corroborate data and
multiple disciplines to broaden one’s understanding of the method and phenomenon
of interest” (McMillan & Schumacher, 2001:408-9). Through the process of
triangulation, the researcher is able to find regularities in the data and in so doing
improve the reliability of the findings. During the data collection process, an
important consideration for me was that the information collected should be
triangulated across the three intended methods to be used, viz.: interviews, lesson
observation and learner written responses to the given tasks.
Figure 7: Graphic representation of triangulation process.
The learners’ written responses to the tasks were compared to their responses in the
interview as well as the alignment to the manner in which the Teacher taught the
lesson, and interviews held with learners. The interview, as a data collection tool,
corroborated the results obtained by learners through analysis of learners’ responses
Interviews with learners
Lesson observations
Learner responses to written tasks
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to the task and observations of how the teacher taught. The nature of developing an
appropriate interview schedule and fears of researchers’ bias resulted in more time
and effort than the other two data collection tools.
The three sources of data are closely intertwined [since they are informed by the same
research questions] and all three informed emerging patterns, results and conclusions.
The triangulation of the three sources is explained in depth in the next chapter. I will
now proceed to discuss how each data collection tool was developed, the intended
data to be gathered and the reliability of the instrument used.
3.2.2 Interviews
Structure and flexibility In order to understand how grade 12 learners reason when solving geometric riders, it
was necessary for me to gather in-depth information from those learners. The learners
were expected to complete the given tasks (see Appendix 2) on their own. The
purposes for the learners to complete the tasks are two fold. Firstly, the strategies,
which learners employ to solve geometric problems, can be ascertained. Secondly,
using the learner’s responses to the task I would then be able to identify the learners to
be interviewed. Thus the research instrument to be used had to be structured yet
flexible. Structure would assist in the gathering of information on pre-determined
topics that the researcher regards as valuable to the study. Flexibility on the other
hand, would aid in capturing the experiences of learners when solving geometric
riders by allowing the researcher to explore and probe unanticipated responses.
Cohen and Manion (1991) are of the opinion that an unstructured interview allows the
interviewer greater flexibility and freedom than a structured interview. They go on to
argue that although the research purpose shapes the question asked, their content,
order and phrasing is left to the researcher’s discretion. In a like manner, in semi-
structured interviews and the interview guided approach (McMillan & Schumacher,
2001) topics are pre-selected in advance, but flexibility is afforded to the researcher in
that the researcher “decides the sequence and wording of the questions during the
interview” (McMillan & Schumacher, 2001: 444).
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The use of an unstructured interview to collect data, in the hands of a skilled
researcher may “produce a wealth of valuable data” (Bell, 1987: 72), but would be
difficult for a novice researcher to implement. The adoption of a semi-structured
intensive interview guide approach thus seemed feasible to obtain qualitative data
from the five (5) grade 12 learners’ from a class of approximately 24 learners
involved in the study, to explore and interpret their understanding of Euclidean
Geometry. The next aspect of selecting the research instrument was its suitability to
generate appropriate data within a qualitative framework.
Interviews as a suitable collection tool in a qualitative framework
The intention of the case study was to provide a qualitative analysis of the manner in
which grade 12 learners’ reason when solving geometric problems. Factors such as
how geometry was taught, learners’ perception about geometry, and the milieu of the
school are all subtle yet important considerations that have an impact on learners
understanding of Euclidean Geometry. The choice of interviews was based on how
best information about these issues could be generated within a qualitative
framework. Concepts about interviews are wide and varied. However, I have
identified the following aspects as appropriate for the use of interviews in a
qualitative environment.
The interview is a data-collection tool, which is widely used, in qualitative research.
The interview is often described as a goal – directed conversation (Macmillan &
Schumacher, 2001; Bell, 1987; Marshall and Rossman, 1994). In an interview the
researcher’s primary goal is to elicit “certain information from the respondent” (Bell,
1987:70); which the researcher regards as important to the research study undertaken.
Qualitative researchers pride themselves in discovering and portraying the multi-tired
views of the case studied and that the interview is the main artery to multitiered
views.
As the research concerns eliciting the “multiple strategies” learners’ employed when
solving geometric problems, in an in-depth manner, the appropriateness of interviews
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to a qualitative framework was not difficult to establish. Elsewhere in this chapter,
some advantages of interviews over questionnaires are listed to demonstrate its
usefulness as a qualitative research tool.
Limitations of the use of interviews as a research tool
Interviews as a data collection tool have certain limitations and weaknesses. It is
difficult to generalize research findings to a wider population as the sample sizes are
normally small and this adversely impacts on the researchers’ validity. In general the
interviewing procedure is not standard and the manner in which questions are based
would differ from interviewer to interviewer. The lack of standardization during the
data collection process, makes interviewing difficult to replicate successfully. Intense
interviewing is prone to interviewer bias (Bell, 1987). The flexibility of the researcher
in formulating questions and probing issues is a potential source of bias. The quality
of the data collected relies heavily on the skills of the interviewer. The lack of
standard processes in the analysis of data can lead to opposing interpretations from a
single body of data gathered.
In qualitative research it is often difficult to overcome the above listed limitations of
interviews. It is significant to acknowledge and at best minimize their influence on the
research process. Regarding bias Bell (1987), notes that its best to “acknowledge the
fact that bias can creep in than to eliminate it together” (page 73). The limitations
listed above need to be viewed in the content of the proposed, general research design.
It should be noted that the limitations above are not all applicable to the case study-
design or to the study undertaken in this report. Generalization of results is not the
focus of case studies in which sample sizes are small deliberately. The purpose of this
study is not to replicate the research design, since each case study is unique, but to
extend the results and findings to learners in similar environments. Bell (1987) cites
Bassy (1981), in which the latter maintains that in case studies, “ The relatability of a
case study is more important than its generalization”.
Cohen and Manion (1991), maintain that validity and researcher bias in interviews are
closely related. They (Cohen and Manion, 1991), go on to suggest that the most
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practical manner to enhance the researcher’s validity is to minimize the amount of
bias as far as possible. They argue that the sources of bias are the characteristics of the
interviewer, the respondents and the substantive content of the questions. Cohen and
Manion (1991) suggest that one of the ways to reduce researcher bias is to have
multiple (different) interviewers but this would be costly and time-consuming.
McMillan and Schumacher (2001), provides us with another view to limit researcher
bias, i.e. that if the interview is done correctly, “it does not matter who the interviewer
is; any number of different interviewers would obtain the same results” (p. 268).
Furthermore, in case studies, researcher bias is best restricted by obtaining actual
quotations as well as accurate records from the participants. To further limit potential
sources of biases, participants should be afforded the opportunity to check records
before the researcher begins to analyse the data gathered.
Bell (1987) states that interviewing in the hands of a capable researcher has potential
to generate “a wealth of valuable data, but such interviews require a great deal of
expertise to control and a great deal of time to analyse” (p. 72).
An unfortunate reality of using interviews is that it is time consuming to prepare,
conduct and analyse and the researcher needs to take cognisance of this before it is
selected as a research tool. The final consideration selecting interviews as a research
tool was its advantages over other data collection tools like questionnaires.
In a qualitative study like this one, the researcher is often interested in aspects, which
are deeply buried in the minds of participants. Thus, in order to reach beyond the
physical reach of the participants, researchers normally use either questionnaires or
interviews.
In this study personal interviews were selected as the preferred mode of collecting
data. Personal interviews are superior to questionnaires because they afford the
following benefits to the researcher: it affords the researcher the opportunity to ask
structured and open ended questions; responses obtained can be probed if there is a
need for such clarification; interactions by participants for clarity and establishing a
personal rapport with the participants involved.
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3.2.3 Summary of advice about designing and using i nterviews
Interviews are frequently used in qualitative research designs as an instrument to
collect data. However, it is also one of the “frequently misunderstood” (Leedy,
1993:192), research tool. The following advice is a summary to be shared between
researchers if researchers would like to conduct successful interviews with the aim of
collecting relevant data. The suggestions are clustered according to the following
categories:
• The initial planning phase;
• The formulation of questions;
• The pilot study;
• The conducting of the interviews; and
• The analyses of data.
In the initial planning phase, it is important for the researcher to do the following:
demarcate the area to be explored during the interview and use this as a guide when
formulating the questions; decide what you need to know and why you need to know
it (Bell, 1987).
When formulating questions, it is suggested that the researcher focuses on the
sequence of the order in which the questions are to be asked (McMillan and
Schumacher, 2001); devise probing questions; pose the same questions to all
participants to ensure comparability of results (Fraenkel and Wallen, 1993); questions
should be worded clearly and unambiguously (Leedy, 1993; Bell, 1989);a mix of
open-ended and direct questions should be used to allow for greater respondent
participation (Thompson, 1978).
In the piloting phase of the study, the researcher should conduct preliminary
interviews with a select few participants who display similar characteristics as the
participants in the main study of the research. By so doing, major shortcomings with
the interview protocol can be identified early on in the study and rectified (Bell,
1987).
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When conducting the actual interview, the researcher should state the focus and
purpose of the study upfront (McMillan and Schumacher, 2001); demonstrate
flexibility by shifting quickly between ideas (Posner and Gertzog, 1982); enhance the
participants self-esteem by making positive remarks to their responses (Bell, 1987;
Osborne and Gilbert, 1980); encourage participants to ask clarity seeking questions
during the interview process. The researcher should not dominate the interview
sessions by talking excessively unnecessarily- instead the researcher should afford the
participants ample time to air their views without any hindrance (Bell, 1987; Posner
and Gertzog, 1982).
When analysing the data the researcher should strive to analyse and interpret the
collected data in terms of the objectives of the research study (Cohen and Manion,
1991); respect the anonymity of the participants as information gathered should be
treated in a highly confidential manner (Bell, 1989); limit their own biases and
personal prejudices towards the study (Bell, 1989; Cohen and Manion, 1991); classify
open-ended questions into sub-categories and these sub-categories need to be verified
by knowledgeable experts (Cohen and Manion, 1991). The researcher should guard
against accepting responses at face value, but should instead validate responses by the
processes of triangulation (Cohen and Manion, 1991; Schumacher and McMillan,
2001).
According to McMillan and Schumacher (2001), the process of effective interviewing
“depends on efficient probing and sequencing of questions” (p. 448). To realise the
goal of effective interviewing, McMillan and Schumacher (2001), have offered the
following guidelines:
• Interview probes should be used to elicit detailed information, further
explanations and classification of responses;
• The researcher needs to articulate the purpose and focus of his/her research
from the outset;
• There should be a semi-structured ordering of questions that would allow for
flexibility to obtain adequate data;
• Demographic questions should either be dealt with throughout the interview
session or in the concluding section of the interview session;
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• Questions of a complex, controversial or difficult nature should be catered for
during the middle or tail-end of the interview session.
Figure 8 below, indicates the steps to follow when designing, implementing and
analysing an interview research instrument in a qualitative study.
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Figure 8: Flowchart for interviews as a research instruments (Chetty, 2003:41)
As stated earlier, elsewhere, the purpose of this study is to gather information through
an in-depth qualitative study about learners’ reasoning ability when solving geometric
riders. Information gathered will be based on learners’ reasoning abilities to solve
geometric riders involving cyclic quadrilaterals and tangent theorems. The
Interview
design
Research purpose
Theoretical basis
Specific
objectiv
Focus questions
Question format
Question response
Data analysis
Pilot studies
Interview schedule
Collect data
15-30 min
Follow interview schedule
15-30 min
Get respondents assent
30 sec
Explain recording process
60 sec
Explain nature of interview
60 sec
Conducting interviews
Coding Patterns
Analyse, interpret and check with participants
Subcategories Categories
Analysis of data
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information so gathered, is intended to benefit educators when teaching Euclidean
Geometry to high (secondary) school learners’.
Another aim of the study is to translate broad research goals into more detailed and
specific research objectives. Hence, the need to prepare the interview schedules.
According to Cohen and Manion (1991), this means that research objectives are to be
translated into questions that would form the core of the interview. Cohen and Manion
(1991), and McMillan and Schumacher (2001), suggest that before the actual
questions are prepared, a thought needs to be spared for the format of the questions as
well as the possible responses to the questions. In the interview the researcher used
both open-ended questions as well as choice selection questions where more direct
answers were required. This is in keeping with the balance of structure and flexibility
alluded to earlier.
According to Frankel and Wallen (1993); open- ended questions in interviews have a
number of advantages. These advantages include the following:
• they are flexible;
• they allow the interviewer to probe respondents for more detail;
• they assist in clearing any misunderstandings; and
• they encourage cooperation and help establish rapport between the researcher
and the respondents.
In the interview protocol presented several open-ended questions are included to
establish how learners’ reason when solving geometric riders. Some choice related
questions were included to maintain focus of the interview so that respondents do not
talk aimlessly and non-stop.
Cohen and Manion (1991) argues that the type of information sought will frame the
response mode as well as the way in which the data collected will be analysed. In the
interview protocol the response modes are a mixture of a task based activity to be
completed by learners (see end of this Chapter for tasks). The data generated was of a
nominal type. The advantage of generating nominal type of data is that it minimises
the bias effect while increasing the flexibility (Cohen and Manion, 1991). However, a
disadvantage of this type of data is that it becomes more difficult to code.
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Following the preparation of the interview schedule was the setting up and conducting
of the interviews. Permission, and on site selection of participants was obtained prior
to conducting the interviews. Bell (1987), suggests that besides a sense of “common
sense and good manners” (p.75), the following courtesies need to be also adhered to
by the researcher:
• Introduce yourself as well as the purpose of your study;
• Respondents should be made to feel at ease;
• The manner in which responses are to be recorded need to be explained to the
participants;
• If recording devices (tape recorder, video recorder, etc.) are to be used, get
participants consent before you actually conduct the interview;
• The interviewer needs to abide by the interview schedule, though not
religiously- there should be room for flexibility.
In addition to the above, if the researcher makes any commitments (promises, etc) to
the participants, then such promises need to be fulfilled. Researchers’ should take
Bell’s (1987) words to heart and remember them “take care not to promise too much”
(p.76).
The final stage in the interview process, once the data has been collected, involves
coding and scoring. Cohen and Manion (1991), regard the process of coding as the
translation of question responses into specific categories for data analysis. In this
study, summaries were made after the interviews or through audio recording of
participants’ responses, which were then catalogued according to pre-determined
categories. In the interview protocol listed, responses were post-coded and matched to
the pre-determined classes in the content analysis. Learners’ responses were then
rank-scored to ascertain the frequency of particular responses occurring. Finally, the
data was analysed and interpreted according to the objectives of the research study
(Cohen and Manion, 1991).
It should be remembered that these stages and processes were not to be adhered to
religiously- they are flexible and subject to change- should the need arise. The
qualitative researcher, by continuously checking and reflecting on what he/she
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planned provides valuable insights for his/her research questions. It is however,
important for a researcher to plan effectively each step of the interview that he/she
will follow in eliciting information from the research question(s). Similar
developmental strategies were applied to the instruments for the document analysis
and the lesson observation.
3.2.4 Document analysis Document analysis as its name suggests can be appropriately used to analyse past as
well as present records of the participants involved in the study. Fraenkel and Wallen
(1993), believe that a “person’s or group’s conscious and unconscious beliefs,
attitudes, values, and ideas are often revealed in the documents they produce” (p.389).
As stated earlier, information collected from lesson observation session and the
interviews was triangulated with information obtained from learners’ responses to the
task-based activities. Personal documents, such as learners’ exercise (work) books;
educator’s lesson plans, etc are a rich source of valuable information that the
researcher can have access if needed.
3.2.5 Participant observation of lessons
Participant observation is commonly used in case studies as it allow the researcher to
“actually participate in the situation or setting they are observing” (Fraenkel and
Wallen, 1993:390). According to Fraenkel and Wallen (1993), as well as Borg and
Gall (1983), the researcher can assume one of two roles when participating in lesson
observations. Firstly, the researcher can be fully immersed in the situation in which
case his true identity is shielded from the rest of the group; and secondly, the
researcher’s participation is partial, meaning that the researcher acts as an observer
but is also allowed to participate fully with the group to establish rapport and develop
a better understanding of the group’s dynamics (i.e. how the group functions, etc.) and
relationships. In this study the researcher assumed the latter role.
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Since the aim of this case study was not an intervention, but exploratory and
interpretative of conditions as they unfolded, there was little need for me to become
fully immersed in the group’s functioning. To ascertain how learners reasoned when
solving geometric riders, it was necessary to observe learners in practice during the
study. An agreed upon time schedule for classroom observations and feedback was
established with the educator.
Two modes of recording data were employed for the classroom observation, viz. field
notes and video recordings. Field notes were descriptive and informal in nature and
selectively used in accordance with the broad categories related to learners’ reasoning
strategies employed when solving geometric riders. Video recordings were not used in
any significant manner.
3.2.6 Learner-based tasks used The questions selected for the task were typical examination type questions, which the
learners may have encountered previously. The questions were well-suited for both
higher and standard grade learners doing mathematics at grade 12 levels.
Whilst learners are not usually asked multiple choice questions in an examination, these questions have been designed to gain a peep into the learners thought processes when solving geometric problems. The use of multiple choice questions instead of routine examination type questions was motivated by the fact the in multiple choice questions the answer “lies in front of the pupil” (Daly, 1995). To each of the questions four alternatives are provided, of which only one is a valid response to the question. The challenge for the learner is that the additional three alternatives also appear to be valid responses to the given question. The questions posed were set or compiled by “experienced teachers who, over the past number of years, have been members of teams of writers, trained to design multiple-choice questions” (Daly, 1995). Accompanying each question, below is a brief description of what the question entails as well as possible routes learners could embark upon to arrive at a valid response to each question.
3.2.7 The task and the administration thereof The tasks (see below) that were administered were based on cyclic quadrilaterals and
tangent theorems. The tasks consisted of eight (8) multiple questions (Section A) and
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one open ended type of question (Section B). In the main study learners were asked
not to answer task 8 in Section A because of limited time. The tasks were first piloted
with a different group of grade 12 learners at the same school. McMillan and
Schumacher (2001) argues that conducting a pilot study affords the researcher the
opportunity to “Check for clarity, ambiguity in sentences, time for completion,
directions, and any problems that may have been experienced”(p.185), before the
instruments could be used in the main study.
Participants for the main study were selected on the basis of the school’s grade 12
results in Mathematics. It is common knowledge that learners’ performance in
mathematics in comparison to other subjects is below par. Schools whose average was
within a 5% range of the District’s (Tshwane South) 2003 average were identified as
potential sites of implementation of the research instrument. Consideration of
distance, travelling time, and accessibility to learners, educators and school were all
factors to be considered before the final selection was made. The latter constraints
resulted in a school in close proximity of the researcher’s home to be selected.
It was important for the researcher to pilot the main data collection instrument, before
actual implementation in the main study. However, due to unsuitable time frames and
the reluctance of the School Management at one possible site, the pilot and main
study were conducted at the same site, but with different groups of grade 12 learners’.
Table 9: Comparison between the pilot and main study. Characteristic Pilot Study Main Study
Gender composition of learners Heterogeneous Heterogeneous
Ability group Mainly standard grade learners Both higher and standard grade
learners
Learner performance Poor to average Average to good
Learning environment Enhances learning Enhances learning
Educator involved Same educator Same educator
The above table (Table 9) illustrates the similarities and differences that prevailed
between learners in the pilot study and the main study. The same educator taught both
groups of learners. The learners’ in the pilot study were mainly standard grade
learners’, whilst the learners’ in the main study were from both higher grade and
standard grade. Both groups were well represented in terms of gender- both groups
- 66 -
had boy-learners and girl-learners in them. As a result of the similarities prevalent in
Table 10, as well as the teachers’ understanding of the need to improve learners
understanding of Euclidean Geometry, it was possible to pilot the main research tool
and expect to obtain data which is reliable and meaningful to the main study.
The pilot study of the learner task was used to check on the following:
• User friendliness to the learners;
• Appropriate language used,
• Layout of diagrams and its aesthetic appeal,
• Ease of marking (scoring) of tasks and the analysis thereof; and
• Searching for and identifying patterns to the learner’s responses.
The pilot study assisted in the refining of diagrams used as well as the terminology
used.
A key aspect of the pilot study was the kind of data generated through the use of the
data collection instruments. Meaningful data in this context related to how learners
responded to the given tasks and the geometric reasons which they advanced in
support of their responses. The data collected indicated that some alternatives had to
be refined and in some cases diagrams as well as the given information had to be
refined to address the research questions adequately. The interview schedule was
expanded to include the following items from the initial three items:
1) Learners’ background, attitude to Euclidean Geometry (original);
2) Learners’ perception about Geometry (original);
3) Learners’ understanding of proof (added),
4) Learners’ “tools” used to solve geometric riders (added), and
5) Learners’ understanding of theorems (original).
These five items corresponded with the emerging patterns observed through learner’s
responses to the tasks, interviews and classroom observation.
As data from the pilot study was analysed it became clear to the researcher that the
manner in which learners responded to the tasks given could be catalogued under
these emerging patterns (numbered 1-5 above). These patterns would be a reliable
- 67 -
source of information as they provided rich data about and answers to the following
overarching research questions for the study viz:
1) How do learners (grade 12 learners’) begin to solve or write a proof to a
given rider?
2) What knowledge and skills do grade 12 learners’ employ in order to solve
geometric riders?
In this study, these emerging patterns of the pilot study evolved into key aspects
against which data in the main study was analysed and key findings identified.
3.3 The main study
3.3.1 Historical data about the educator and learn ers’ concerned.
The educator has a three year Senior Primary Teacher’s Diploma after grade 12
(M+3). The educator has taught in a secondary school for the past 20 years. The
educator has taught all grades in a secondary school, i.e. grades 8-12. Despite the
teacher’s qualifications, the teacher has consistently achieved a 100% pass rate at
grade 12 level since teaching grade 12’s in 1998. The teacher has been twice
nominated for the Gauteng Mathematics, Science and Technology Teacher of the year
award for consistently producing excellent results at grade 12 levels.
The learners who participated in the main study were ‘doing well’ in mathematics
according to the teacher. Out of the class group of 27 learners’ two learners achieved
an E symbol as a year mark and two achieved an F symbol as a year mark. The
remaining learners in the class obtained symbols A-D as a year mark (see Table 12).
Furthermore 16 of the learners entered to write their end of grade 12-year exams on
the higher grade and the remaining 11 entered as standard grade candidates. The class
had 18 female learners and 9 male learners. Of the 27 learners, only 24 learners
completed the tasks- the other three were absent on the day the tasks were given.
- 68 -
INSTRUCTIONS:
1. Thank you for agreeing to be part of my research project. 2. This is NOT a test. 3. The task is divided into two (2) sections.
• SECTION A: Multiple choice items. • SECTION B: Open ended question.
4. Answer ALL the questions from both SECTION A and SECTION B. 5. Show all working details, where necessary, on the blank page
opposite each question.
NAME OF STUDENT:
- 69 -
SECTION A:
• Each of the questions has four (4) alternatives to them. • Circle the letter which you think is the most appropriate response
to the question. • Provide a motivation for your choice to each question on the blank
page opposite each question
1. In the given diagram, A, B and C are points on the circumference of the circle. E is the centre of the circle. AD is a tangent to the circle at A. AC and AB are equal chords of the
circle. 030B ADÙ
= . The size of ......CEBÙ
=
A) 300 B) 1200
C) 900 D) 600
C
E
A
B
D
In order for the learner to be able to solve this rider it is anticipated that the learner will :
• Identify the angle between the tangent and the chord (i.e. angle BAD = 300 ) as been equal to angle BCD;
• Next use the fact that AC = AB to deduce the size of angle BAC =1200;
• Construct an angle on the major arc of CB which will be equal 600;
• Use this fact then to calculate the angle CEB , using the angle at centre theorem ,i.e . angle CEB = 1200
- 70 -
2. Points A, B, D, E and F lie on the circumference of a circle.
0EBC = 80Ù
and 0AEB =35Ù
. The magnitude of EDBÙ
is .………
A) 1250 B) 1000
C) 1350
D) 450
G
BA
F
E
D
C
In order for the learner to be able to solve this rider it is anticipated that the learner will:
• Identify two unique cyclic quadrilaterals, viz. EABD and FABE;
• Next use the fact that angle BEA = angle BFA = 350 (angles subtended by chord AB);
• Next they should deduce that angle EBA = AFE = 800 (exterior angle of cyclic quad. Equal to interior opposite angle);
• By performing some basic arithmetic i.e. angle AFE – angle AFG = 450 and hence angle EDB = 1800 – 450 = 1350 (opp.angles of a cyclic quad are supplementary)
- 71 -
3. In the given figure QUTS is a cyclic quadrilateral. PQR is a tangent
to the circle at Q. TS = TU ; SU = SQ and TP ║ SQ. If SQR = xÙ
, which angle is not equal to x .
A) 1PÙ
B) 1QÙ
C) 1UÙ
D) 1 2S SÙ Ù
+
3
2
1
2
1
2 1
1
U
T
Q
R
P
S
In order for the learner to be able to solve this rider it is anticipated that the learner will:
• Identify the angle between the tangent and the chord, i.e. angle SQR = x and then the angle in the alternate segment, i.e. angle SUQ;
• Next, using the fact that SU = SQ ,learners can deduce that angle SUQ = angle SQU= x(angles opposite equal sides ;
• Next using the fact that TP║SQ ,learners can identify corresponding angles, viz. angle TPQ and angle SQR;
• Based on the above reasoning then the correct option to choose would be option C.
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4. PQR is a tangent to the circle at Q. QU is parallel to RST. UQ = US
and UT = TS. If xRQS∧∧∧∧
==== , then the value of x is …………..
A) 450 B) 360 C) Cannot be determined D) 720.
T
S
Q
R
P
U
This rider may appear to look like the previous one –but it differs in that the learner is asked to determine the numerical vale of the variable x. In order for the learner to be able to solve this rider it is anticipated that the learner will:
• Identify the angle between the tangent and the chord, i.e. angle RQS = x and then the angle in the alternate segment, i.e. angle SUQ;
• Next, using the fact that SU = UQ ,learners can deduce that angle SUQ = angle SQU= x(angles opposite equal sides ;
• Next using the fact that TS║UQ ,learners can identify a pair of alternate angles, viz. angle QUS and angle UST;
• Learners can then express angle UTS as 1800 – 2x (remaining angle in ∆UTS); • Deduce that angle UQS = 2x (opp.angles of a
cyclic quad. are supplementary); • By working with the sum of interior angles
of ∆UQS, learners can then deduce that x = 360, which is option B.
- 73 -
5. KLMO is a cyclic quadrilateral of a circle with centre P.
0NMO 120∧∧∧∧
==== and 0LPM 80∧∧∧∧
==== . Which one of the following statements is FALSE?
A) OM = ML
B) MK L and K ML∧ ∧∧ ∧∧ ∧∧ ∧
are complementary angles.
C) M PL OK M∧ ∧∧ ∧∧ ∧∧ ∧
====
D) 4 OMK PKL∧ ∧∧ ∧∧ ∧∧ ∧
====
M
P
L
O
K
N
This rider is different from the previous one in that the learners are asked to identify the FALSE statement. In this task the term “complementary angles” is used. It is not included to “derail” learners but rather to assess whether they are familiar with terms such as “complementary angles” , which mean that angles add up to 900. In order for the learners to be able to solve this rider it is anticipated that the learner will:
• Identify the exterior angle of the cyclic quadrilateral angle NMO = angle OPL (exterior angle of cyclic quad. = interior opp. angle);
• Next, using the fact that P is the centre of the circle then PM = PL = KP = PL (radii);
• Using the above fact, learners can then deduce that angle PLM = angle PML = 500. Similarly it can be shown that angle PKL = angle PLK = 400
• Base don the above reasoning learners can then deduce that option A is definitely FALSE.
- 74 -
6. XYZ is a common tangent to the two circles. With respect to the given diagram which of the given statements is TRUE?
A) ∆ BYC /// ∆ DYE
B) BC BYDE BE
====
C) ED ║ XZ
D) BY CYEY YD
====
DE
B C
Y ZX
This rider is different from the previous one in that the learners are asked to identify the TRUE statement. Whilst all four alternatives look TRUE at first glance, this may be misleading.. In order for the learners to be able to solve this rider it is anticipated that the learner will:
• Identify the angle between the tangent and the chord relative to the larger circle ,i.e. angle XYB equal to angle YCB (angle in alternate segment),and to angle EDY in the smaller circle. Similarly in relation to the smaller circle the angle between the tangent and chord angle ZYD is equal to angle YED in the smaller circle and to angle YBC in the larger circle;
• Based on the above learners will deduce BC // ED because the re appears to be a pair of corresponding angles equal, which would then make option D the correct one.
• Options A , B and C are incorrect because: Option the order of the triangles are not written properly; option B is not valid deduction and option C looks like something they have dealt with previously.
- 75 -
7. AE is a tangent to the circle at A. CDE is a straight line. 0DAE 30
∧∧∧∧==== and 0APD 120
∧∧∧∧==== . The size of ADP
∧∧∧∧is
____________________
A) 600
B) 900
C) Cannot be determined. D) 300
1200
300
P
D
C
A
F
E
B
• This diagram was deliberately distorted by the researcher. The learners were not aware of the fact that the diagram was distorted. The reason for the distortion was to ascertain whether learners could visualise that the measures for the given angles are not applicable to the given diagram. It is anticipated that learners would go for option C because of the nature of the diagram and the values given. This rider wants to explore the “myth” that learners at school believe that there should be a solution to every given rider, especially when numeric values are given.
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8. WPQ is a tangent to the circle at P, which meets SR produced in Q.
TR║PQ. If ST = 4,5 and SQ = 18 then the length of SP is________________
A) 2 B) 22,5
C) 2 2(4,5) 18++++ D) 9
R
P QW
T
S
LEARNERS WERE ASKED NOT TO ANSWER THIS RIDER BECAUSE OF TIME CONSTRAINTS.
- 77 -
SECTION B Show all working details on the blank page opposite this one.
1.In the given diagram O is the centre of the circle. VOX∧∧∧∧
= 700. Points U, V and X lie on the circle. Calculate the sizes of the following:
1.1) U∧∧∧∧
1.2) 2V∧∧∧∧
1.3) XWV∧∧∧∧
1
32 1
432
1
O
X
Y
W
UV
• For 1.1. learners had to apply the angle at centre
theorem to calculate U∧∧∧∧
; • Using the above result learners could calculate
2V∧∧∧∧
using the fact the angle V2 = angle X3 (angles opp. equal sides) . thus angle V2 = 550;
• Considering the limited information given, its not possible to calculate the size of angle XWV. However learners were able to come up with a variety of “solutions” to this question.
- 78 -
Chapter 4: Data Analysis and findings
4.1 Analysis of learners responses
Coding process
To safeguard the identity of the participants (learners and teacher) involved, codes
were used when referring to either party. The teacher was referred to as Teacher,
whilst the learners’ responses were coded L1 –L24. The “L” refers to learner and the
numbers 1-24 is the order in which the researcher collected the response sheets from
the learners. The responses of the 24 learners were then catagorised and analysed
according to the following broad categories.
Table 10: Categories of learner’s responses.
Cat
egor
y
num
ber Category
C1 Correct response with valid reasons and working shown.
C2 Correct response with some valid reasons and working shown.
C3 Correct response with no valid reasons and working shown
C4 Incorrect response with valid reasons and working shown.
C5 Incorrect response with some valid reasons and working shown
C6 Incorrect response with no valid reasons and working shown
C7 No response but some attempt made to solve question
C8 No response, with no attempt to solve question.
Table 11 (below), shows each learner’s response to the given task. Table 11 also
provides some historical information regarding each learner who completed the task.
The above categories were formulated prior to the learners responding to the tasks. Based on my experience as a maker at grade 12 level as well as my classroom teaching experience, the above categorise were developed. Often learners would provide a correct response for instance without showing any working or reasoning as to how he/she arrived at a solution.
- 79 -
Table 11: Learners responses according to categories outlined in Table 10 SECTION A SECTION B
Lear
ners
Gen
der
Leve
l
Yea
r m
ark
1 2 3 4 5 6 7 8 1.1 1.2 1.3
L1 F H C C2 C1 C1 C5 C1 C5 C1 C1 C1 C3
L2 F S A C4 C1 C5 C5 C1 C1 C1 C4 C1 C1
L3 M S A C6 C6 C6 C5 C1 C5 C6 C1 C1 C2
L4 M H C C1 C1 C1 C5 C1 C1 C5 C1 C1 C5
L5 F H B C1 C1 C2 C1 C5 C1 C1 C1 C1 C6
L6 F H F C2 C5 C5 C5 C5 C3 C5 C1 C1 C7
L7 F H A C1 C2 C2 C5 C1 C1 C1 C1 C1 C4
L8 F S E C7 C6 C5 C5 C6 C5 C7 C1 C3 C6
L9 F H C C1 C7 C5 C5 C6 C6 C7 C1 C6 C8
L10 F H D C6 C5 C1 C5 C1 C6 C1 C1 C1 C5
L11 F H C C2 C3 C5 C5 C2 C5 C5 C1 C1 C3
L12 F S B C6 C6 C5 C5 C5 C5 C5 C1 C1 C8
L13 M H C C1 C1 C1 C1 C1 C5 C1 C1 C1 C6
L14 M S F C3 C5 C6 C1 C2 C3 C2 C1 C1 C6
L15 M H B C2 C1 C1 C1 C1 C5 C2 C1 C1 C6
L16 M S D C2 C2 C3 C5 C2 C2 C5 C1 C1 C8
L17 M S E C2 C2 C2 C5 C5 C2 C2 C1 C1 C5
L18 M S B C2 C1 C5 C6 C6 C6 C3 C1 C1 C8
L19 F H D C5 C2 C6 C6 C5 C6 C6 C1 C1 C8
L20 F S A C1 C1 C5 C6 C1 C1 C1 C1 C1 C6
L21 F H B C1 C1 C5 C6 C1 C1 C1 C1 C1 C6
L22 F H B C1 C7 C1 C6 C6 C1 C1 C1 C1 C8
L23 F H C C1 C1 C6 C6 C2 C1 C6 C1 C1 C8
L24 F S D C2 C7 C5 C6 C6 C5 C6
NO
RE
SP
ON
SE
C8 C8 C8
2Codes used in above table (See footnote)
The next stage in the classification process was to classify how the learners responded
to each of the questions in Section A of the activity. Table 12, below reflects how the
learners responded to each of the questions in Section A.
2 Codes used in Table 11: Level refers to the grade on which learners have entered for the end of year exam (i.e. either higher or standard grade), F = female learner , M = male learner; H = higher grade ; S = Standard grade
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Table 12: Learner’s response to tasks in Section A Number of learners who selected… as a response
Question No. A B C D No Response Total
1 3
(12, 5%)
16
(66, 67%)
0
(0%)
3
(12, 5%)
2
(8, 33%)
24
(100%)
2 0
(0%)
2
(8, 33%)
14
(58, 33%)
3
(12, 5%)
5
(20, 83%)
24
(100%)
3 3
(12, 5%)
0
(0%)
11
(45, 83%)
10
(41, 67%)
0
(0%)
24
(100%)
4 6
(25%)
4
(16, 67%)
14
(58, 33%)
0
(0%)
0
(0%)
24
(100%)
5 14
(58, 33%)
6
(25%)
1
(4, 17%)
3
(12, 5%)
0
(0%)
24
(100%)
6 9
(37, 5%)
0
(0%)
3
(12, 5%)
12
(50%)
0
(0%)
24
(100%)
7 0
(0%)
2
(8, 33%)
12
(50%)
7
(29, 17%)
3
(12, 5%)
24
(100%)
8 NO RESPONSE BY ALL LEARNERS 24 3Cells shaded
The above table (Table 12) indicates at a glance the number of learners who selected
each of the given responses (A-D) as their choice as well those who have not provided
any response to the given task. Whilst the tables 11-13 are rich quantitatively, they
need to be unpacked further so that it can become intelligible to the general
readership.
Question by question analysis of learners’ response s
What follows is a selection of some learner’s responses to the questions in both
Section A and Section B of the task. The sample responses are from learners who
have provided both correct and incorrect responses. The rationale was to provide a
spectrum of responses as well as learners’ reasoning ability when solving the tasks.
3 Shaded in cells in Table 12 refer to the correct solution(s) for a particular question in Section A. For 3 both options C and D are valid options- hence both are shaded .
- 81 -
Question 1 (Section A)
Table 13: A summary of learners’ responses to Question 1 Number of learners who selected… as a response
A B C D NO
Response
Total
No. of
learners
3 16 0 3 2 24
% of
learners
12,5% 66,67% 0 12,5% 8,3% 100%
The above table is a summary of how learners responded to the given question.
One of the learners, L24 supplied the following proof4 to the given task.
Table 14: L24’s response to Question 1 Line Statement Reason
1 ∧E = 180o -30o
Opp angles of cyclic quad
2 = 150o
3 ∧F = 300o (angle at centre = 2 X angle at circle)
4 360o – 300o =
2
60o
5 ∧E= 30o
4 The learner (and others) has not supplied their proofs in tabular form. The researcher has done so because of convenience only.
1. In the given diagram (on the right), A, B and C are points on the circumference of the circle. E is the centre of the circle. AD is a tangent to the circle at A. AC and AB are equal chords
of the circle. DAB∧
=300. The size of
BEC∧
is……………. A) 300 B) 1200
C) 900
C
E
A
B
D
- 82 -
When viewing L24’s response closer the following aspects are worth noting:
The learner has indicated BAC∧
, instead DAB∧
as 30o;
a) The learner proceeds to calculate ∧E using that fact that ECAB is acyclic
quadrilateral (according to the learner);
b) The learner concludes (see calculations above) that ∧E = 30o, but circles
option B (the correct option) as a response to task 1.
The learner’s initial response (see line 2 above), of 150o is closer to the 120o than the
other options supplied. Furthermore, both 120o and 150o are three digit numbers and
120o is 30o less than 150o, hence the choice of 120o, despite the working reflecting
something else.
Furthermore, L24 has erroneously identified ECAB as a cyclic quadrilateral, because
three (C, A and B) of the four vertices lie on the circle. This is a distortion of the
definition of a cyclic quadrilateral which states that a “cyclic quadrilateral is a
quadrilateral of which the vertices lie on a circle” (Laridon et al, 1995: 277). What
needs to be emphasized in this definition is that ALL four vertices must lie on a
circle. L24’s misrepresentation of the definition of a cyclic quadrilateral can be
explained in terms of Movshovitz-Hadar, Inbar and Zaslavsky’s (1986) notion of
“distortions of the consequent” (p.34), in which the original condition is maintained or
the original condition is slightly modified to fit in with the learners’ view at the time.
Below is L24’s diagram used to solve the given task.
- 83 -
Figure 9: Learner 24’s diagram used. Note markings on the diagram which informed her choice of answer.
Another learner, L2 (see Figure 10 below) interprets the given information correctly,
i.e. identifies the angle between the tangent and the chord equal to the angle in the
alternate segment, radii (CE =EB) as well as angles opposite the equal sides, viz
E∧C B = E
∧B C. Once the learner calculated (determined) the size of C
∧AB as 120o the
learner erroneously identified ECAB as a cyclic quadrilateral and hence deduced that
C∧E B = 60o. In this case the learner used (applied) the theorem “if a quadrilateral is
- 84 -
cyclic then the opposite angles are supplementary” (Laridon et al, 1995:277),
incorrectly. CEBA is not a cyclic quadrilateral, since only three of the four points lie
on the circle. Like L24, L2 has not fully grasped the understanding of what a cyclic
quadrilateral is.
Figure 10: The diagram used by L2 to answer Question 1. Note the markings used on the diagram.
Another learner, L3, proceeded to determine the size of C∧E B using the tangent-chord
theorem. However, C∧E B is not the angle in the alternate segment, although AB is a
chord of the given circle. Once again Movshovitz-Hadar et al’s (1986) notion of
“distortions of the consequent” (p.34) is applicable to this learner’s thinking.
- 85 -
Question 2 (Section A)
Table 15: A summary of learners’ response to Question 2 Number of learners who selected…… as a response
A B C D NO
Response
Total
No.
learners
0 2 14 3 5 24
% of
learners
0% 8,33% 58,33% 12,5% 20,83% 100%
In this question the learner’s understanding of cyclic quadrilateral theorems was
assessed. From the above table we can see that 58, 33% (14/24) of the learners were
able to provide the correct response to the question posed, while 41, 67% (10/24) of
the learners were unable to do so.
Learners who selected option D as a response argued along similar lines to L8 whose
proof is given below.
Points A, B, D, E and F lie on the circumference of a
circle. CBE∧
= 800 and BEA∧
= 350. The magnitude
of ∧
DBE is .………
A) 1250 B) 1000
C) 1350
D) 450
G
BA
F
E
D
C
- 86 -
Table 16: L 8’s response to Question 2 of Section A Line
no.
Statement Reason
1 1
∧F + 2
∧F = 80o
Ext angle of cyclic
quadrilateral
2 2
∧F = 45o
The proof, which L8 provided above, assisted L8 to determine the size of BDE∧
.
However, L8 then applied properties of other quadrilaterals like squares or rhombus
in which the opposite angles are equal to determine the size of BDE∧
. The learner
failed to realize that the properties of squares or rhombus would not as a rule of thumb
apply to all quadrilaterals.
L10, whilst attempting to come up with a solution, failed to do so. Judging from Table
11, L10 is an “average” Mathematics higher-grade student, who is expected to cope
with the type of riders provided in the task. However, L 10’s response “I don’t know
how 2 do it!!! Sorry- I gave up!!”(emphasis added) seems to suggest that he lacks
confidence and the necessary skills and know-how to do the task. The learner is able
to identify and apply theorems relating to cyclic quadrilaterals (for example angles in
the same segment), but lacks the geometric eye to differentiate between an interior
and exterior angle of a cyclic quadrilateral.
In this question, learners were given five points, which lie, on the circle. These five
points are then joined (see given sketch above) and three cyclic quadrilaterals are
formed. Learners’ were expected to indicate the given information on the sketch,
which would have enhanced their ability to solve the given rider successfully. By
successfully indicating the given information the learner’s “geometric eye” (Godfrey
as cited in Fujita and Jones, 2002:384) would have come into play. The learner’s
ability to “see” a solution is enhanced when the geometric eye is well developed, i.e.
when learners are able to visualize how “geometrical properties detach themselves
from a figure” (Godfrey in Fujita and Jones, 2002: 385).
- 87 -
Question 3 (Section A)
Table 17: Summary of learners’ responses to Question 3
Number of learners who selected…… as a response
A B C D NO
Response
Total
No.
learners
3 0 11 10 0 24
% of
learners
12,5% 0% 45,83% 41,67% 0% 100%
In this question both options C and D were regarded, as correct responses to the given
task.
In this question learners had to apply their knowledge of previously acquired
geometric concepts to identify angle(s) not equal to a given angle. Whilst the task
may have been straightforward, learners’ incorrectly read SU = UQ instead of
SU = SQ as was given. L6 for instance committed this type of error. As a result of
this incorrect labelling the learner arrives at a correct deduction response to the given
question.
In the given figure QUTS is a cyclic quadrilateral. PQR is a tangent to the circle at Q. TS = TU ; SU = SQ and TP ║ SQ. If
RQS∧
= x , which angle is not equal to x .
A) ∧
1P
B) 1Q∧
C 1U∧
D) 21 SS∧∧
+
3
2
1
2
1
2 1
1
U
T
Q
R
P
S
- 88 -
Figure 11: The diagram L 6’s used to solve Question 3. Note the markings on the diagram made by the learner.
Question 4 (Section A)
PQR is a tangent to the circle at Q. QU is parallel to RST. UQ = US and UT = TS.
If xR Q S∧∧∧∧
==== , then the value of x is
A) 450 B) 360
C) Cannot be determined
D) 720.
T
S
Q
R
P
U
- 89 -
Table 18: Summary of learners’ response to Question 4 Number of learners who selected…… as a response
A B C D No
Response
Total
No.
learners
6 4 14 0 0 24
% of
learners
25% 16,67% 58,33% 0% 0% 100%
In this question the learners knowledge on tangents and cyclic quadrilaterals was
integrated with basic geometric facts established in previous grades. In this question
learners’ were expected to determine the numeric value of the variable x. From the
above table we can see that 25% (6/24) of learners opted for option A and 58, 3%
(14/24) opted for option C as their respective responses to the question.
Learners were expected to formulate a linear equation of the type 5p = 10 and hence
solve for p. However the learners may be familiar in doing that when working with
algebraic equations but they are not always expected to do likewise when solving
geometric problems. A typical response to the given question, which demonstrates the
learner’s lack of ability to formulate an equation, is supplied below.
- 90 -
Table 19: The proof provided by L12 to Question 4 Line
no.
Statement Reason
1 ∧Q = x
2 x = U Tan-chord theorem
3 But: U = S = x Angles opp equal sides
4 U = S = x Alt. angles QU // RST
5 T = x Ext angle = int.opp angle
6 Q = 180o – 2x Opp angles of cyclic quad
supp
7 180o = 2x
8 2x = 180o
9 x = 90o ÷2
10 = 45o
The above proof is an attempt by L12 to answer the given question. In line 5, L12 has
incorrectly identified R∧QS as an exterior angle of a cyclic quadrilateral QUTS. In
terms of the van Hiele model of thinking, L12 is functioning at an inappropriate level
for the current grade. Learners at senior secondary level (i.e. at grade 10-12 level) are
expected to be functioning at least at level 3 (i.e. informal deduction). However, L12
seems to be functioning at level 1, since he has not yet developed the competency to
make sense of the relationships between properties, and “interrelationships between
figures are still not seen, and definitions are not yet understood” (Crowley, 1987:2).
Besides L 12’s inability to make sense of theorems, L12 attempts to solve the problem
using his algebraic knowledge of equations to solve the given rider. L12 recalls
solving equations of the type 5p – 10 = 0 and tries to use that knowledge to solve the
given task. L12 knows that line 6 (∧Q = 180o – 2x) needs to be rearranged to look like
5p –10 = 0. Line 7 resembles this type of mental activity the learner is engaged in.
The mathematical rules used in lines 8 -9 are true, although line 7 is mathematically
flawed. The learner’s response in line 9, whilst correct, undergoes a further “division”
- 91 -
process because the learner’s response of 90o is not on the list of answers provided.
Hence, the further division by 2 to match the response of 450, which is option A.
Another learner, L10 concluded that the value of x cannot be determined because
“NO values are given”, hence he selected option C.
For this learner in order to determine the value of x some other definite value(s) had
to be provided. This would have facilitated the determining of the value of the
variable x. Another learner, L2, who also selected option C, provided the following
response.
Table 20: A response to Question 4 by L2 Line
No.
Statement Reason
1 180o - (180o – 2x + x +x)
2 = 180o
3 = 0
This response by one of the better achieving learners in the class (See Table 11)
indicates that learners’ only possess procedural knowledge on how to solve geometric
riders. They lack the ability to apply knowledge gained to new situations in order to
arrive at a valid and plausible solution.
Both L2’s and L12’s responses can be explained in terms of Bernstein’s (1996)
recognition and realization rules. Cooper and Dunne (2000), explain Bernstein’s twin
concepts of recognition and realization as follows: “Recognition rules, ‘at the level of
the acquirer’ are the means by ‘which individuals are able to recognize the specialty
of the context that they are in’. Realization rules allow the production of ‘legitimate
text’” (Bernstein, 1996 in Cooper & Dunne, 2000:48). Whilst both learners were able
to recognize the given information relating to tangent, and cyclic quadrilaterals, etc.,
they failed to produce the “legitimate text” (Cooper & Dunne, 2000:48), i.e.
appropriate response.
- 92 -
Question 5 (Section A)
Table 21: Summary of learners’ responses to Question 5 Number of learners who selected…… as a response
A B C D No
Response
Total
No.
learners
14 6 1 3 0 24
% of
learners
58,33% 25% 4,2% 12,5% 0% 100%
In this question learners had to identify using the given information, the statement that
is FALSE. From the above table, we can deduce that more than 50% (14/24) of the
learners were able to select the correct alternative, i.e. option A (OM = ML).
Learners’ could have “guessed” the response to be A , which would have been an
educated “guess”. Learners’ could have measured OM and ML and found that they
are not equal and deduce accordingly that that statement is FALSE.
Those learners’ who had selected option B, could have done so after supplying a
proof similar to the one supplied by L8 below:
KLMO is a cyclic quadrilateral of a circle with
centre P. 0N M O 1 2 0∧∧∧∧
==== and
0L P M 8 0∧∧∧∧
==== . Which one of the following
statements is FALSE?
A) OM = ML
B) M K L a n d K M L∧ ∧∧ ∧∧ ∧∧ ∧
are
complementary angles.
C) M P L O K M∧ ∧∧ ∧∧ ∧∧ ∧
====
D)O M K P K L∧ ∧∧ ∧∧ ∧∧ ∧
====
M
P
L
O
K
N
- 93 -
Table 22: A response to Question 5 by L8 Line
no.
Statement Reason
1 L
∧P M = 80o
Given
2 ∧K = 40o Angle at centre = 2X angle
at circle
3 L
∧P M = 100o Angles on a straight line
4 L
∧M P = 50o Angle at centre = 2X angle
at circle
Learners’ such as L8, who have opted for option B, have not mastered the linguistic
demands associated with Euclidean Geometry yet. The concept of complementary
angles though not frequently encountered refers to angles whose sum adds up to 90o.
It is a concept they have encountered during their study of Trigonometry mainly.
Question 6 (Section A)
Table 23: Summary of learners’ responses to Question 6 Number of learners who selected…… as a response
A B C D No
Response
Total
No. learners 9 0 3 12 0 24
% of
learners
37,5% 0% 12,5% 50% 0% 100%
XYZ is a common tangent to the two circles. With respect to the given diagram which of the given statements is TRUE? A) ∆ BYC /// ∆ DYE B) B C B Y
D E B E====
C) ED ║ XZ
D) B Y C Y
E Y Y D====
DE
B C
Y ZX
- 94 -
In this question learners had to apply their knowledge of tangent theorems to similar
triangles. Although 50% (12/24) of the learners were able to select the correct
response, an equal number of learners also selected the incorrect response. Whilst
learners in both groups were able to identify the angle(s) between the tangent and
chord and the angle in the alternate segment- many of them made the wrong
conclusions.
Learners who selected option A failed to adhere to the basic principle when dealing
with similar triangles. The order in which the vertices of the triangles are written is of
paramount importance. Textbook authors such as Gonin et al (1997) stress this point.
When naming similar triangles, “the letters indicating corresponding angles should be
written in the same order for all triangles”(p.358) (emphasis added). Learners failure
to realize that although ∆ DYE and ∆ EYD refer to the same figure, the order in which
the vertices appear do not correspond to the corresponding vertices in ∆ BYC. Thus
the correct order should be ∆ BYC /// ∆ EYD and from this the ratio statement
YD
CY
EY
BY = follows. Attention to order in naming triangles is important, something
which a large number of learners have not yet mastered.
Question 7(Section A)
AE is a tangent to the circle at A. CDE is a
straight line. 0D A E 3 0∧∧∧∧
==== and
0A P D 1 2 0∧∧∧∧
==== . The size of A D P∧∧∧∧
is
____________________ A) 600
B) 900
C) Cannot be determined. D) 300
1200
300
P
D
C
A
F
E
B
- 95 -
Table 24: Summary of learners’ responses to Question 7 Number of learners who selected…… as a response
A B C D No
Response
Total
No.
learners
0 2 12 7 3 24
% of
learners
0% 8,33% 50% 29.2% 12,5% 100%
The diagram to this question was deliberately distorted to gain an insight into
learners’ spatial understanding. The above table reflects that while 50% of learners
were able to visualize that the given information does not match the diagram and
hence a solution is not possible, an equal number of learners, because of previous
experiences, are determined to find a solution at all costs.
Learners’ lack of spatial sense more often than not hinders their ability to visualize
forms given to them and because of a poor spatial sense make incorrect judgments.
Fujita and Jones (2002) cite Atiyah (2000), who writes that:
“spatial intuition or spatial perception is an enormously powerful tool and that is why
geometry is actually such a powerful part of mathematics- not only for things that re
obviously geometrical, but even for things that are not. We try to put them into geometrical
form because that enables us to use our intuition. Our intuition is our most powerful tool..”
(p.386).
- 96 -
SECTION B
The tasks in Section B were open ended in that no possible values for:
1.1) ∧
U ;
1.2) 2
∧V and
1.3) X∧
W V is provided.
The responses to the task 1.3 in Section B, can be categorised into the following:
i) A remark “impossible to find out” (response by L2); “cannot find”
(response by L23);
ii) No response or attempt to solve the task; and
iii) Some attempt made by the learners to solve the given task.
For tasks 1.1 and 1.2 all learners’ were able to provide legitimate responses to the
given tasks. However for task 1.3 some interesting responses were provided. Below is
an example of one such “unusual” solution.
L3 provided the following response to task 1.3.
In the given diagram O is the centre of the circle.
VOX∧∧∧∧
= 700. Points U, V and X lie on the circle. Calculate the sizes of the following:
1.1) U∧∧∧∧
1.2) 2V∧∧∧∧
1.3) X W V
∧∧∧∧
1
32 1
432
1
O
X
Y
W
UV
- 97 -
Table 25: L 3’s response to 1.3 of Section B
Line Statement Reason
1 4
∧X = 1
∧O + 2
∧V
Ext angle of triangle
2 4
∧X = 700 +550
3 4
∧X = 1250
4 1
∧V = 1
∧O + 3
∧X
Ext angle of triangle
5 = 700 + 550
6 1
∧V = 1250
W is impossible to calculate since the sum of 1
∧V and 4
∧X is
greater than 1800 and the angles of a triangle add up to 1800.
In the above proof, L3 regards 4∧X and 1
∧V as exterior angles of triangle OXV. One
can see that L3 does not have an understanding of the concept of an exterior angle of
a triangle. L3 may not have developed the “schema” (Chinnappan, 1998:202) of
triangles adequately. Chinnappan (1998), defines the concept of “schema” as “a
cluster of knowledge that contains information about core concepts, the relations
between these concepts and knowledge about how and when to use these concepts”
(p.202). Considering Chinnappan’s notion of “schema”, L3 lacks insight on how and
when to use the knowledge on exterior angles of a triangle. However, despite L3’s
error in Lines 1 and 4 above, he arrives at a valid conclusion, viz. “the sum of 1
∧V +
4
∧X is greater than 1800, hence
∧W cannot be determined”, demonstrates that L3’s
schema on the sum of angles of a triangle is better anchored than the schema on
exterior angles of a triangle.
In the given task (1.3), neither XO nor OV is produced to warrant the learner’s
justification for the use of the exterior angle of a triangle theorem. Although the
learners’ reasoning is flawed, the learner is able to arrive at a valid conclusion, viz.
that it is impossible to calculate∧
W .
- 98 -
Another learner, L10 provided the following solution to task 1.3.
Table 26: L10’s response to 1.3 of Section B Line Statement Reason
1 X
∧W V = 1800 - 700 Properties of a kite
2 X
∧W V = 110
This is a clear assumption!
I have no idea how 2 do it. I
just don’t see a solution!!!
(Learners remark to statement in line 2)
Considering the appearance of the figure OXWV, L10 assumes that the figure is a
representation of a kite. The learner then proceeds to use the properties of a kite to
calculate the size of X∧
W V. In this case, L10 draws on the schema of kites to solve
the given problem. However, not enough information has been supplied for the
learner to conclude positively that OXWV is a kite. Like L10 mentions, an
assumption was made to enable him/her to arrive at a solution- because this is what
school geometry is all about- there has to be a solution to every rider.
L11, in an attempt to solve the task (1.3), provided a solution based on the appearance
of the figure. L11’s response was “Look impossible, there are no cyclic quadrilateral
and neither a tangent”. Furthermore, the learner (L11) expects, because of previous
encounters with similar riders, to be provided with a cyclic quadrilateral and/ or a
tangent in order to arrive at a solution to the task.
Learners such as L4, L5 and L3 who attempted to solve the task, based their responses
on the theorems which they have studied thus far. L4 and L5, for instance based their
solution on the angle at centre theorem to determine the size of∧
W . The point O is
given as the centre of the circle passing through the points U, X and V- but the learner
expands the circle to include the point W as well. Based on this type of flawed
reasoning, L4 provides the following proof.
- 99 -
Table 27: L4’s response to 1.3 of Section B ∧
2O = 3600 - 700 (angles around a point)
= 2900
X∧
W V = 1450.
From the preceding discussions, it has become apparent that learners employ different
strategies in order to solve a given geometric problem. The strategies employed by the
learners’ are often based on amongst others previous encounters with similar
problems, the visual appearance of a given figure, and the incorrect application of (a)
theorem(s).
From the responses arriving at a solution ,valid or otherwise meant to the learner that
he/she had provided a proof to the given rider. For the learner a proof meant writing a
statement with a reason alongside it. A proof had to be in a two column table format
for the learner with a reason attached alongside each statement. The learners never
considered the mathematical value of their proof to a given rider.
Whilst I had intended to employ the interview method as a data collection tool, I had
to disregard that strategy, as the data derived from that exercise was not adding value
to my study. The responses to the task, I believe were able to provide me with a much
richer source of data then the interviews did.
The learners’ strategies to solve a given problem can be clustered according to the
following categories:
• Incorrect use of theorems to solve a given problem;
• Responses to a problem are based on the visual appearance of a figure;
• Learners’ responses to a problem is based on their previous experience with a
similar problem; and
• Learners’ view of proof to a given problem.
Each of the above categories will be explored in some detail in the next chapter.
- 100 -
Chapter 5: Discussion of findings
From the preceding analysis of learners’ responses to the task in both Sections A and
B, the following observations are made regarding learners’ responses to the given
task:
• Incorrect use of theorems to solve a given task (rider);
• Responses to tasks are based on the visual appearance of a figure;
• Learners “force” a solution to a given task (rider), although no solution is
possible at times; and
• Learners demonstrated different views of proof.
5.1 How learners use theorems to solve a given problem.
The tasks given to the learners are typical riders, which they have encountered during
their study of Euclidean Geometry. The theorems or tools which learners could draw
on, were based on tangents and cyclic quadrilaterals theorems integrated with
theorems based on triangles, as well as circle geometry theorems such as angle at the
centre of a circle.
The geometric riders consisted of four common geometric concepts regularly
encountered by learners: circle; tangent, triangle and quadrilateral. The riders were
developed by having these concepts “integrated in a manner which demanded that
the solver to recognize a component as serving more than one function” (Chinnappan,
1998:205). For instance, in Question 4 (see Appendix 2), the side QR needs to be
identified as (i) a straight line, (ii) a tangent to the circle at Q; and (iii) a side of the
quadrilateral RQUS. This recognition of one part of the figure playing multiple roles
constitutes an important part in the modelling process before students are able to
recruit appropriate theorems in order to generate new information. For instance the
recognition of QR as a tangent could aid learners to infer that R∧QS = Q
∧U S = x
(tangent-chord theorem). Furthermore, the identification of QU// RST could result in
learners using alternate angles, i.e. Q∧
U S = U∧ST and the added fact that UT = TS
- 101 -
could result in learners inferring that S∧
U T = T∧SU = x (base angles of an isosceles
triangle) in triangle UTS. Having identified the angles equal to x and trying to express
the unknown angles in terms of x, learners’ could have calculated the numerical value
of x. However, 80% (20/24) learners’ provided option C (i.e. cannot be determined)
as a response. This high “negative response” immediately begs the question WHY?
In a research study undertaken in Israel by Movshovitz-Hadar, Inbar and Zaslavsky
(1986), about learner’s responses to exam type questions, they were astonished at the
number of student-invented variations of theorems. A possible contributing factor that
hampers learners’ in providing an accurate mathematical proof may be learner’s lack
of understanding of a theorem and then misapplying it. Movshovitz-Hadar et al
(1986) refer to this as “distortion of theorems” (p.26). An example of the phenomenon
of distortion of theorems is illustrated in the following example.
Table 28: A learner’s response to illustrate the phenomenon of “distortion of theorems”
∧
4X = 1
∧O + 2
∧V (ext < of triangle)
∧
4X = 700 + 550
∧
4X = 1250
∧
1V = 1
∧O + 3
∧X (ext < of a triangle)
= 700 + 550
∧
1V = 1250
∴∧
W is impossible to calculate since the sum of ∧
1V + ∧
4X is greater than 1800 and the
angles of a triangle add up to 1800.
In the above example, L3 identified ∧
4X and ∧
1V as exterior angles of triangle OXV.
What is true is the fact that ∧
4X and ∧
1V are outside (exterior) angles of triangle OXV.
However, the theorem related to the exterior angle of a triangle cannot be applied to
the given rider in this case. The learner, L3, arrives at a valid conclusion based on
incorrect or flawed reasoning. The above example demonstrates what Weber (2002)
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has identified as a student’s “lack of understanding of a theorem or a concept and
systematically misapply it” (p. 102). Furthermore, the above example demonstrates
this particular learner’s lack of appropriate mental schemas that would assist the
learner to recruit the appropriate theorems that would result in a valid solution.
Whilst this study is limited in its number of participants, the tasks used were able to
demonstrate that some learners were just not able to leave their starting blocks. For
instance in:
• Question 1 : 8,33% (2/24) learners did not attempt the task;
• Question 2 : 20,83% (5/24) learners did not attempt the task;
• Question 7: 12, 5% (3/24) learners did not attempt the task.
Learners’ lack of attempting the tasks cited above would suggest that the learners’
inability to provide an attempted solution is as a result of “they reach an impasse
where they simply do not know what to do” (Weber, 2002:102).
5.2 Learners responses are based on the visual appearance
of a given figure According to Monaghan (2000),
“The conceptual distance that students must cover to move from the stage of
recognizing such gross visual features of shapes as straightness or length to more
abstract concepts such as parallel ness or perpendicularity is far greater than the
mere difference in vocabulary might suggest. Students very early on are able to
recognize and distinguish shapes. What is less clear is the basis on which they
make such distinctions” (p.184).
From research undertaken by Monaghan (2000), in which secondary school learners
had to differentiate between different quadrilaterals, it emerged from the study that
the learners used properties of one kind of rectangle for all rectangles. Monaghan
(2000) refers to Hasegawa (1997) who provides the following comment in this regard:
“The prototype is a result of our visual-perceptual limitations which affect the
identification ability of individuals, and individuals use the protypical example as a
model in their judgments of other instances” (Hasegawa, 1997 in Monaghan, 2000:187).
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The above view of Hasegawa (1997) is well illustrated in the manner in which
learners responded to the tasks given. To demonstrate this assertion, learners
responses to some of the tasks will be dealt with below.
In task 1 of Section A (see Appendix 2), learner’s knowledge on:
1. tangent-chord theorem;
2. angle at centre theorem;
3. equal chord properties; and
4. properties of isosceles triangles,
is being probed. The diagram, though, is not a typical prototype diagram found in
your textbook, in terms of the above knowledge foci. In terms of the van Hiele model
of geometric reasoning, secondary school learners should have surpassed the visual
stage (level 0) of the model. The Revised National Curriculum Statements (Grades R-
9) (DoE,2002), in their assessment standards for geometry suggest that learners
exiting the senior phase(i.e. the end of grade 9) of schooling, should be operating at
least at level 2 of the van Hiele model of reasoning. Learners at the senior phase
should be able to “describe and represent the characteristics and relationships between
2-D shapes and 3-D objects in a variety of orientations” (p.6). Thus, one would expect
that a grade 12 learner would be in a position to identify the key constituent parts of
the given diagram and then recruit the necessary theorems to successfully solve the
given rider. However, whilst not a generalisation, learners’ employed cyclic
quadrilateral theorems (e.g.: opposite angles of a cyclic quadrilateral are
supplementary, i.e. C∧AD + C
∧E B = 1800) to solve the given task- despite there being
no cyclic quadrilateral present in the diagram.
Another example where learners made incorrect judgments based on the given
diagram is evident in task 6 of Section A (see Appendix 2).
In this task, based on previous encounters with similar examples, 37,5% (9/24) of the
learners’ deduced that triangle BYC /// triangle DYE and a further 12,5% (3/24)
learners’ deduced that ED//XZ. Both these responses are typical questions which
learners are expected to answer (see Appendix 3).
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Jones (undated), maintains that learners’ “previous experience and the visual image”
of a figure shapes the manner in which a learner would solve or attempt to solve a
geometrical rider. Jones, cites Fischbein(1987) in which the latter asserts that
“Experience is a fundamental factor in shaping intuitions”(p.82); and visualization is
the primary factor “contributing to the production of the effect of immediacy”(p.82).
The mental modalities, which learners’ recruit in an attempt to solve geometrical
riders is thus shaped by the following two assumptions:
1. the visual appearance of the figure; and
2. learners previous engagement (experiences); with similar type of
problems.
Thus, the learners intuitive reasoning plays a significant part when formulating a
formal argument to a given geometrical rider.
5.3 Learners “force” a solution to a given task although no
solution is possible at times
The classroom tasks given to learners are traditionally designed by text book authors
to yield a solution. However, some of the tasks in the activity sheet, were designed
with the aim of not yielding a solution. Task 7 in Section A and task 1.3 in Section B
are two such examples. For task 7, 37, 5% (9/24) of the learners opted for options B
(90) and D (30), whilst 12, 5% (3/24) did not provide a solution and the remaining
50% (12/24) selected option C- the correct solution. It is thus note- worthy to explore
how the 37, 5% (9/24) learners arrived at option B or D in order to get insights into
their reasoning.
The following learners L3, L6; L4 and L23 opted for option D (300) – using different
theorems related to cyclic quadrilaterals and tangent theorems and generic properties
of quadrilaterals. L3 and L23 (see Appendix 3), first used the tan-chord theorem
to show that E∧AD = A
∧B BD = 300 and thereafter a potpourri of angles in the same
segment and assuming that P is the centre of the circle to deduce that A∧D P = 300.
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L 4 (see Appendix 3), after proving that A∧B D = 300, indicated that triangle ABP
is an equilateral triangle (check markings on sketch) - but indicates different
measures for the interior angles of triangle ABP. This demonstrates to me that the
learner has not yet fully grasped the properties of an equilateral triangle. A similar
discrepancy prevails in triangle PCD of the same figure.
Both sets of solutions cited above are indicative of the learners’ yearning to provide a
numeric solution to a given geometric problem when numeric values are given for
angle sizes. This “forced” type of solution indicates that the type of problems learners
have been exposed to always resulted in a definite solution.
Similarly in 1.3 of Section B (see Appendix 2), learners were asked to calculate the
value of X∧
W V. Once again, based on the visual appearance of the given diagram and
the learners previous engagements with similar problems, learners’ (e.g. L4 ; L5)
assumed that O is the centre of the circle passing through UXWV. Based on this
assumption, learners proceeded to calculate the size of X∧
W V using cyclic
quadrilateral and angle at centre theorems. Other learners, such as L10, L 18 and L 21
assumed that XW is a tangent to the circle at X and proceeded to use the tangent-
chord theorem or tangent perpendicular to the radius to determine the size of X∧
W V.
Both sets of responses cited above, as a result of the visual appearance of the
diagrams and their intuition, learners recruited inappropriate theorems to assist them
to solve the given riders. Furthermore, both sets of responses highlighted above
suggest that learners (regardless of their ability level) have not yet grasped the
theorems they are working with sufficiently. Faulty assumptions made on the basis of
the visual appearance of a given diagram suggest that learners have not as yet
developed the “geometrical eye” (Godfrey, 1910, cited in Fujita and Jones, 2002:385)
for detail. The geometrical eye, as defined by Godfrey (1910), relates to the learners’
ability to see “geometrical properties detach themselves from a figure” (Godfrey, as
cited in Fujita and Jones, 2002:385).
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In addition to developing and nurturing the geometrical eye, learners need to develop
and nurture an additional skill to be able to solve mathematical problems in general
and geometrical problems in particular, i.e. “geometrical power” (Godfrey, 1910 in
Fujita and Jones, 2002:388). The concept of geometrical power relates to the learners
“power of seeing geometrical properties detach themselves from a figure” (Godfrey,
in Fujita and Jones, 2002:388). As an example of Godfrey’s notion of the
“geometrical eye” consider task 6 of Section A (see Appendix 2).
To be able to identify the correct statement, learners had to see (visualize) that
triangles BYC and EYD (note the order of the triangles) are likely to be similar.
Learners were able to deduce that triangle BYC /// triangle DYE (option A), but failed
to take note of the order of the second triangle- thus eliminating option A. Similarly,
learners failed to see that ED is not parallel to XZ, but ED is parallel to BC. Fujitsa
and Jones (2002) cite a study undertaken by Nakashini (1987), with 87 Japanese
learners aged between 14-15 years in which they had to prove AZ = BY if triangle
XYZ is an isosceles triangle. Although 75% (65/87) of the learners’ were able to
provide a correct response to the problem, there were others 25% (22/87), who were
unable to “see” the solution.
To be able to successfully solve geometrical riders, one should also have developed a
well-trained “geometrical eye” which will assist in arriving at a valid solution. A
geometrical eye will not just develop overnight – it’s a process which requires the
intervention and support of all educators teaching geometry- from the foundation
phase educator right through to the educator who teachers at grade 12 level and
beyond.
“There must be a good foundation of practical work, and recourse to practical and
experimental illustration wherever this can be introduced naturally into the later
theoretical course. Only in this way can the average boy [sic] develop what I will
call the ‘geometrical eye’” (Godfrey, 1910 in Fujitsa and Jones, 2002:388).
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Chapter 6: Summary, recommendations and conclusion
6.1 Summary and findings
6.1.1 Overview
The aim of this study was to explore, grade 12 learners’ understanding of Euclidean
Geometry with special reference to cyclic quadrilateral and tangent theorems. The
primary focus of this study was to investigate what cognitive tools learners recruit in
order to solve geometrical riders.
In Chapter two, the van Hiele model of geometric thought was discussed and
provided the theoretical framework for collecting, analyzing, interpreting and
reporting grade 12 learners’ understanding of cyclic quadrilateral and tangent
theorems. The findings by examiners of grade 12 exams as well as the MST (2003)
Report provided justification for this study. Grade 12 learners’ poor performance in
exams in geometrical riders involving cyclic quadrilateral and tangent theorems meant
that it was important for this study to analyze this group of grade 12 learners’
understanding of the mentioned theorems.
In Chapter three, the choice of a case study as an appropriate research tool was
explained. Using a case study allowed the researcher to explore strategies of how
learners’ reason, discover important questions to ask during the interview and try to
understand learners’ thinking processes. The choice of interviews and tasks in the
case study design allowed for a rich description and analysis of data.
In Chapter four, learners’ responses were analyzed, coded and categorised as
described in Section 4 .1 ( Table 10), see page 77.
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Based on the above, the following key aspects were identified in an attempt to
understand learners’ reasoning skills, namely:
Key
Asp
ect
No.
Aspect
1 Inappropriate use of theorems to solve a given rider
2 Visual appearance of figure plays a role in solving rider
3 Learners “force” a solution to a rider, even when one is not possible
4 Learners view of proof to a rider
Table 29: Key aspects according to which learner’s reasoning skills were categorised. These four aspects facilitated a detailed analysis of learners reasoning ability. In
Chapter five, each of the four key aspects identified in Chapter four were discussed
in more generic terms.
6.1.2 Primary research questions and sub-questions
The primary research questions in this study are:
1. How do grade 12 learners begin to solve a geometric problem?
2. What knowledge and skills do learners recruit in order to solve geometric
problems?
The primary questions were further broken down into the following sub-questions:
1.1 Why did you (the learner) follow a certain route (plan) when solving a given
geometrical problem?
1.2 What type of information was provided either directly or by implication in the
given tasks?
1.3 How do you (the learner) view proofs?
Why did learners follow a certain route (plan) when solving a given geometrical problem? In this study learners’ engagement with similar types of problems and the visual
appearance of the given diagram(s) framed the manner in which learners approached a
given problem. The tools (theorems, definitions, axioms, etc.), which learners used,
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were primarily based on the visual appearance of the figure. Whilst learners were able
to identify the theorems to be used, they often came up with their own contrived
version of a particular theorem such that it corresponded with the given task.
Learners vision of geometric concepts appears to be blurred at times. Whilst some
aspects are obvious from the given information, at other times what learners may see
(or read) from the given information may be “blatantly wrong things” (Dreyfuss,
1999:105). However, we need to guard against learners who are able to arrive at
correct conclusions using visual reasoning as “correctness of the answer is not the
issue, certainly not the main issue” (Dreyfuss, 1999:105).
What type of information was provided either directly or by implication in the given diagram(s) and/ or text?
The given diagrams were preceded by a textual description of the task. Using the
description provided, learners were expected to transfer the given information (data)
onto the diagram. By so doing learners would then begin to recruit (identify) the
appropriate theorems, definitions, etc. that would facilitate the successful solution to
the given task. However, this intention was not always realized.
Often learners were not able to “see” the solution to the problem, although the given
information was indicated on the diagram(s). As a result of this impediment, learners
recruited inappropriate theorems, definitions, etc. and thus arrived at non-valid
conclusions. At times the solutions provided were not aligned with the learner’s
markings on the diagrams.
Based on their previous engagements with similar type of problems, learners made
assumptions; such as XVW is a tangent (see task 1 of Section B), although this was
not given directly or indirectly to the learners. As a result of this faulty assumption
learners used inappropriate theorems, definitions, etc to solve the given task(s).
Similar assumptions were made regarding task 7 of Section A. In that figure (task 7 of
Section A), learners assumed that P is the centre of the circle- because it looked like
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the centre of the circle, and thus used theorems such as angle at the centre of the circle
to solve the given task.
Thus learners failed to read the question carefully before deciding on a route to the
solution. Often learners relied on their “gut-feeling”-based on the appearance of the
figure and their previous encounter with similar problems when solving a given task.
No consideration was given to the facts given –when learners embarked en-route to
solving a task.
Learners view of proof
In this study, for learners, to prove something meant having a neat two-column table-
like proof with reasons alongside each statement. The learners were exposed in the
main to a deductive axiomatic system of proof. Many learners were afraid to venture
out of this protected environment,
“ it is often assumed that students believe that valid proofs may only be constructed by using a
chain of deductive reasoning to connect axioms, definitions, and already established theorems
within a particular axiomatic system”(Martin, McCrone, Bower and Dindyal, 2005:121),
in this case the deductive –axiomatic system.
Research conducted into mathematical proof concludes that a large body of learners’
lack understanding of the nature of proof (see for example, Senk, 1985), and often
they (learners’) don’t reason in a logical, coherent manner. A study conducted by
Senk (1985) in the United States of America, for instance, concludes that as much as
70% of high (secondary) school learners do not understand the proofs they study.
Learners’ often confuse a worked example of a geometrical rider with a proof. The
learners’ often focused on the appearance of a proof (two column proof), than on the
logical, coherent flow of mathematical ideas (content).
Weber (2001), in his research with university students, has identified two generic
characteristics of learners’ difficulties with proof. The first one has already been
alluded to, viz. learners’ lack of a clear idea of what constitutes a proof. The second
and more relevant and pertinent difficulty which his research has uncovered is the
students’ lack of “understanding of a theorem or a concept and systematically
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misapply it” (Weber, 2001:102). Whilst learners were able to identify the theorem to
be used, they often distorted it or misapplied it. More often than not learners’
despondency leads them to throw their arms up in the air because “students often fail
to construct a proof because they reach an impasse where they simply do not know
what to do” (Webber, 2001:102). (emphasis added)
6.2 Implications and recommendations for classroom practice
It has been argued in this research that Euclidean Geometry poses several challenges
to grade 12 learners’ reasoning ability. Adopting an alternative teaching strategy for
Euclidean Geometry will imply that many educators would be removed from their
present comfort zone of presenting geometric theorems as a finished product. In the
context of this study, the following recommendations can be put forward as a means
to enhance learners reasoning ability in terms of Euclidean Geometry.
How to teach Geometry
To begin with, we need to re-look at the manner in which we teach definitions of
geometric concepts to learners. The direct teaching of geometric definitions has come
under the spotlight by mathematicians and mathematics educators’ alike (De Villiers,
1997). De Villiers (1998), for instance quotes Human (1978), in which the latter calls
for a “reconstructive” approach instead of the more regular direct axiomatic-deductive
approach to teaching of definitions. Human (1978), as cited in De Villiers (1998),
differentiates between the two approaches in the following manner:
“With this term (reconstructive), we want to indicate that content is
not directly introduced to pupils (as finished products of mathematical activity), but that the
content is newly reconstructed during teaching in a typical mathematical manner by the teacher
and/or pupils” (p.1).
The pedagogical advantages of employing a reconstructive approach are:
• Its implementation accentuates the meaning of the content;
• It allows the learners to become actively engaged in the
construction of the content.
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Thus employing a reconstructive approach is therefore characterized by not
“presenting content as a finished (pre fabricated) product” (De Villiers, 1998:1), but
instead to focus on the real mathematical activities through which the content is to be
developed.
Researchers, such as Ohtani (1996) (in De Villiers, 1998), for instance have argued
that the provision of definitions by educators is to ensure that there is, amongst others
a uniformity of the definition as understood by all the students; the educator exerts
some kind of control over the learners; and there is no long drawn out debates and
discussions concerning learners definitions. These are all “un-constructive” and out of
sync with the current curriculum reform practices which encourages learners to
• “communicate appropriately by using descriptions in words,
graphs, symbols, tables and diagrams;
• Use mathematical process skills to identify, pose and solve
problems creatively and critically” (DoE, 2003:9).
Thus to enhance learners understanding of geometric definitions, it is necessary to
encourage learners’ to engage in activities that will afford them the opportunities to
develop the requisite skills which the curriculum intends to develop within the
learners.
It is an accepted fact that all learners are not on the same cognitive level in terms of
understanding, ability level, etc. Therefore it is advisable for learners to provide
definitions of concepts, which align with their cognitive level. From our discussion of
the van Hiele levels of understanding, it was noted that learners develop a clear
understanding of definitions from level 3 only. It would thus be futile to attempt to
provide learners with formal definitions of concepts when they are not yet cognitively
receptive to such demands. At grade 8 level for instance, learners are still at van
Hiele levels 1 or 2.It would thus be futile to expect learners at this stage to provide a
formal definition for, say a rectangle. At this stage learners would focus on the visual
aspect of the figure and provide a definition that corresponds with the visual
representation of the figure. A rectangle could be defined in terms of the length of its
sides, and a typical definition could be: “A rectangle has all angles 900 and two long
and two short sides” (De Villiers, 1997:46). Whilst learners at the stage of level 2 of
- 113 -
the van Hiele model of understanding, may provide a long, cumbersome definition of
a rectangle, it should be noted that such a definition would be in accordance to the
learners level of maturity and development. A typical definition at level 2 could be:
“ A rectangle is a quadrilateral with opposite sides parallel and equal, all angles 900,
equal diagonals, half turn symmetry, two axes of symmetry through opposite sides,
two long and two short sides, etc” (De Villiers, 1997:46).
The process of constructing definitions in mathematics should not be seen as a less
important activity than the process of factorization, for instance. To increase learners’
understanding of geometric definitions then, it becomes incumbent on every educator
to engage learners “in the process of defining geometric concepts” (De Villiers,
1998). Learners should not be provided with ready made definitions of concepts such
as cyclic quadrilaterals, tangents, rectangles, etc. By providing learners with such
ready-made definitions, a misconception is created in the learners that there exists
“only one correct definition for each concept” (De Villiers, 1995). Learners are thus
denied the opportunity to search for alternative definitions, in cases where definitions
are presented as items cast in stone.
“Defining concepts accurately in mathematics is certainly not an easy task, and is only developed after
lots of experience and practice. However, the educational experience is worth the trouble, and I would
like to encourage our authors and teachers out there to seriously rethink their treatment of geometry
definitions” (De Villiers, 1995).
It is an established and universally valid fact, that learners’ poor performance in
Geometry can be attributed to, amongst other factors, the pedagogical (teaching)
strategy of the educator, an outdated curriculum and text book authors who
perpetuated the cycle of presenting ready-to use, neatly packaged content (theorems,
definitions, axioms , etc) , which learners are expected to memorize and accept
without question. Martin et al (2005), argues that if learners performance in Geometry
is to improve, then the educator needs to be the catalyst for change in this process. To
this end, Martin et al (2005), assert that
“We conclude that pedagogical choices made by the teacher, as manifested in the teacher’s actions, are
key to the type of classroom environment that is established and, hence, to students’ opportunities
to hone their proof and reasoning skills” (p.95).
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Whilst most traditional geometry courses were product driven, such an approach does
not cohere with the current constructivist approach to teaching and learning, which
advocates that:
1. “ knowledge is not passively received but actively built up by
cognizing subject;
2. the function of cognition is adaptive and serves the organization of
the experiential world, not the discovery of ontological reality”
(Jaworski, 1996:16).
Thus to achieve this goal, we need to encourage learners to become actively engaged
in the construction of knowledge- especially Euclidean Geometry knowledge.
Teach to prove or teach for proof?
The standard or traditional view of proof in Euclidean Geometry has always been one
of verification of the correctness of mathematical statements. However, this limited,
naïve view has come in for strong criticism. De Villiers (1990), for instance argues
that verification should be seen as one part of the five tiered purpose of proof, which
includes:
1. Verification (concerned with the truth of a statement);
2. Explanation (providing insight into why it is true);
3. Systematization (the organization of various results into a
deductive system of axioms, major concepts and theorems);
4. Discovery (the discovery or invention of new results);
5. Communication (the transmission of mathematical
knowledge) (De Villiers, 1990:18).
Traditionally, learners are given predetermined statements to prove. As a result
learners assume then that the statement must be true. For learners to appreciate the
value of proof writing and to engage as mathematicians do, they need to know how
mathematicians use proof. Learners need to be aware that proof is only one aspect in
the process of learning and discovering new mathematics. Mathematicians first make
conjectures, which are based on observations, or hunches (one’s gut-feeling). The
mathematician then tests these observations or hunches before embarking on a formal
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proof for his conjecture. Before a mathematician can formally publish his/her
findings, he/she allows their conjectures to be critiqued by fellow mathematicians
before their conjecture is accepted as true. Mathematics in the National Curriculum
Statement (NCS) for Further Education and Training (FET), strives to develop in
learners the ability to
• “work collaboratively in teams and groups to enhance mathematical
understanding;
• collect, analyze and organize quantitative data to evaluate and critique
conclusions” (DoE, 2003:10),
this coheres with the tasks and responsibilities of a mathematician. As educators, we
need to afford learners the opportunities to evaluate the thinking processes of their
colleagues and their own.
Using De Villiers (1990), view of proof educators and textbook authors should then
design activities (tasks) that would encourage learners to engage in the above
strategies.
Both Piaget and van Hiele have outlined strategies that would enable learners to prove
ideas formally. For Piaget, learners thinking in general progresses from a stage of
“non-reflective and unsystematic, to empirical and finally logical-deductive” (Battista
and Clements, 1995:50). In a similar manner, van Hiele argues that learners geometric
thinking processes, progresses from stages of lower levels
(visual→ descriptive) of thinking to more complex stages (descriptive → abstract→
formal→ deduction → rigour), a process which is labour intensive and time
consuming. The van Hiele model of thinking suggests that the teacher’s instruction
(pedagogy)should aid learners to gradually progress from the lower levels of thinking
before engaging them (learners) in the rigors of “proof-oriented study of
geometry”(Battista and Clements,1995:50). The premature dealing of formal proof
will not aid learners’ understanding of proof; instead such an approach will result in
“students only to attempts at memorization and to confusion about the purpose of
proof” (Battista and Clements, 1995:50).
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Gole (2003) argues that textbooks in the USA present solutions to geometric riders in
a linear manner implying that learners would be able to make sense of such a solution.
That may be a valid approach to simple riders, but such an approach may be less
effective when dealing with more complex problems. An alternative approach would
be “Sherlock Holmes’s backward reasoning” (Gole, 2003:544) type of approach. This
process of backward reasoning allows the learner to search for possible solutions to a
given rider, using the given information and making valid inferences (deductions)
from such information. Consider the following example wherein this process of
backward reasoning is used to solve a given rider.
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Given: Circle O with POS a diameter and ST a tangent.
Prove: WV // ST (Source: Laridon et al, 1988:338)
Figure12: A typical geometric rider which grade 12 learners’ should be able to solve.
In order to prove WV// ST one would have to either prove:
• a pair of alternate angles equal ;or
• a pair of corresponding angles equal; or
• a pair of co-interior angles supplementary.
A quick glance at the above diagram shows that 1
∧W and 1S
∧ are angles in alternate
positions of the straight lines WV and ST. Thus if these angles can be proved to be
equal to each other, then the task would be solved. But how does one prove something
like this by using the information given? The following flow diagram (see next page)
represents a possible path to solving the task.
21
1
1
2
1V
R
TS
O W
Q
P
- 118 -
.
Figure 13: A schematic representation demonstrating the method of “backward reasoning”.
The above figure is intended to demonstrate the thought process involved in using the
strategy of backward reasoning. One begins at one’s destination point and retraces
one’s footsteps to one’s current location before one can begin with the write up of the
solution of a given problem.
GOAL: Prove: WV // TS and
hence 1
∧W = 1S
∧
Start here
)diameterPOS(
RQ 022 90==
∧∧
deduce deduce
QVRW is a cyclic quadrilateral
because
deduce
Proceed this way
BUT
d
educ
e
11
∧∧= WS
Each = ∧
1Q
CONCLUSION
090=∧
WRV
Adj.supp.<s
WRVQ∧∧
=2
Ext. < =int.opp ,’s
11
∧∧= WQ
<’s in same seg
11
∧∧= QS
Tangent-chord theorem
- 119 -
Often one would hear learners’ comment that if only they knew where to start with a
given rider then they would not feel frustrated and despondent. A good starting point
is to start with what you need to prove and then work backwards. By so doing one
limits the element of luck when arriving at a suitable line of argument. This skill of
backward reasoning is neither natural nor commonly used in mathematics today. The
educator’s help may be an essential ingredient in developing and nurturing the skill of
backward reasoning by providing learners’ with sufficient guided examples over an
extended period of time thereby “geometry proofs provide simplified controlled
problem-solving contexts and help cultivate through repetition the mental search
habit” (Gole, 2003:545).
Learning such skills (backward reasoning), can result in both general and practical
rewards for the learner. An important positive spin-off in the development of such
skills is that learners levels of confidence “in searching for solutions to unfamiliar and
possibly over whelming problems by learning how to limit options” (Gole, 2003:545),
increases. If one had to view the skill of backward reasoning as an investment over a
period of time, then one can conclude that the returns on such an investment would be
far greater than the risks attached to it, since once a learner has mastered the skill, it is
highly unlikely for the learner to forget the skill. Furthermore, mastery of such
thinking skills in geometry can also develop in learners’ important hands-on skills to
manage their everyday lives.
Use of interactive technology
In line with seeking workable alternatives to the rigid axiomatic approaches to
Euclidean Geometry, the focus on computer programs such as Geometers’ Sketchpad
is to facilitate and enhance learners’ ability to making and testing of conjectures.
Researchers such as De Villiers have argued strongly in favour of dynamic software
programs such as Geometers’ Sketchpad that would be able to revolutionize
Geometry at all levels. De Villiers, as cited in Yushan, Mji and Wessels (2005),
defends his claim for interactive software as follows “the main advantage of computer
exploration of topics…is that it provides powerful visual images and intuitions that
can contribute to a person’s growing mathematical understanding” (p.17). By
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visualizing a problem, learners are able to have a global picture of the problem to be
solved.
It is well known that the more senses we use in our teaching and learning episodes,
the more we understand and retain knowledge. By using interactive software such as
Geometers’ Sketchpad, educational technology appeals to our senses of sight, sound
and touch. From a constructivist point of view, then learners become “more active
agents in managing and ensuring the success of their education- invariably sustaining
their attention and commitment to mathematics” (Yushan, et al, 2005:18).
Using Sketchpad diagrams for exploration does not only encourage learners to make
conjectures, but can also develop “insight for constructing proof” (Battista and
Clements, 1995:52). For example, in Sketchpad a learner can construct a circle, locate
four points at random on the circle and consider the quadrilateral so formed. If the
size (measures) of the angles is measured, the points assume different positions on the
circle, and the learners would observe that the sum of the opposite angles of such a
quadrilateral approaches 1800.
The Sketchpad demonstration is convincing, since the size of the circle can be
changed and the vertices moved easily. But will our conjecture hold? If so why? If
not, why not?
6.3 Conclusion
It was the intention of this study to make a meaningful contribution to the body of
knowledge related to learners’ understanding of Euclidean Geometry. Educators have
often lamented learner’s poor performance in Geometry- learners having difficulty in
“seeing” a solution or a path to a possible solution; or learners are unable to make
sense of the theorem(s) to be studied. As a result learners become frustrated, de-
motivated and indifferent towards the subject (Geometry), because they felt
incompetent in dealing with it.
Researchers like De Villiers (1997) have argued strongly that geometry is alive and
well. In fact it is experiencing a renaissance in most countries, including South Africa,
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at all levels of education. Recent curriculum changes in South Africa, for instance
demonstrate a marked shift from the traditional approach to geometry. The study of
geometry at grades 10-12 level, for instance, engenders to develop in the learners the
ability to:
• “explore relationships, make and test conjectures….;
• investigate geometric properties ….in order to establish, justify and prove
conjectures;
• use construction and measurement or dynamic software, for exploration and
conjecture” (DoE, 2003:14).
Current curriculum reform initiatives are thus in keeping with changes in the approach
to the teaching of geometry in other parts of the world.
For the curriculum reform initiatives to be of any significance, there need to be a
radical re-look at the teacher education programmes at both pre-service and in-service
level. Most high school educators know a little more geometry than the learners they
are expected to teach- through no fault of theirs. Institutions of higher learning
offering teacher education programmes need to have compulsory courses in Euclidean
and non-Euclidean geometry for both primary and secondary educators if any
meaningful change in learner-performance is to be registered.
Whilst this study focused primarily on learners’ reasoning when solving geometric
problems, I believe that an investigation into:
• how teachers reason when solving geometric problems and when teaching
geometry;
• the relevance of the language used by educators and its suitability to the
learners’ conceptual understanding in geometry;
in terms of the van Hiele model offers one an opportunity then to broaden and deepen
one’s understanding of the model. I believe that such a study would be able to inform
effective teaching of Euclidean Geometry. What I mean is that there is a need to
understand geometry teaching practice at the chalk face: how teachers teach
geometry, how they use language the language of geometry, and to investigate the
extent to which their use of the language of geometry takes into account learners’
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level of development in terms of the van Hiele model. We need to explore further to
the extent to which providing practicing and pre-service educators with opportunities
to engage in activities that “require classifying answers by van Hiele levels” (Feza
and Webb, 2005:45) might contribute to effective practice in geometry.
Reflections on my research journey
After two years of hard work, I have finally reached the goal I set myself two years
earlier. The road to reach this destination was not always smooth. I had to contend
with challenges at home, at work, unfavourable deadlines imposed by lecturers,
employers and family. Each constituency demanded their due on time. Despite these
unfriendly, at times hostile conditions, I was able to come out on the other end
academically enriched and fulfilled. The goal of my study was not only to be able to
write the letters M.Sc behind my name – there was a greater goal. That greater goal
was to assist my colleagues who experience difficulty when teaching Euclidean
Geometry, to teach it in a manner which allows for greater learner participation in the
discipline. Furthermore, the dilemmas we experience regarding Euclidean Geometry
is not restricted to our shores only - it is a universal phenomenon experienced at
varying levels by different countries around the world.
To conclude, I would like to impart some advice to the novice researcher. Firstly,
make sure that if you are married, your spouse has given you his/her consent and
support to continue with your studies. Failing which, cancel your registration
immediately. Secondly, make sure that you have sufficient time assigned for your
studies. Trying to fit your studies into your schedule does not work – instead let all
activities revolve around your studies. Thirdly , select an area or topic, which you
would like to explore which your supervisor also shows a keen interest in.
Alternatively, try to adjust your topic/area of research which fits into your
supervisor’s field of interest. Finally, do not try to do too much – set yourself realistic
and achievable goals, DO NOT TRY TO BE OVER AMBITIOUS , especially if
you are a novice researcher.
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